GCSE Maths: Circle words and isosceles triangles (Higher)
Part of the GCSE Maths circle theorems chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Circle words and isosceles triangles
A circle is made of all the points that are the same distance from one fixed point, the centre. You met the main words in Chapter 18. Circle theorems depend on them, so every word must be exact. A diagram usually uses lots of these words at once, and you need to spot them quickly.
Words to know
- Radius: a straight line from the centre to a point on the circle. All radii of one circle are equal.
- Diameter: a chord through the centre. It is twice the radius, and it is the longest chord.
- Chord: a straight line that joins two points on the circle.
- Arc: a part of the circumference. The shorter one is the minor arc, the longer is the major arc.
- Sector: the region between two radii and an arc (a slice of pizza).
- Segment: the region between a chord and an arc. A chord cuts a circle into two segments.
- Tangent: a straight line that touches the circle at exactly one point, the point of contact.
Two radii always make an isosceles triangle. Every radius is the same length. So if A and B are on a circle with centre O, then OA = OB and triangle OAB is isosceles. The two equal angles (the base angles) are at A and B, which are the points on the circle. This one idea is used in almost every proof in the chapter. Remember from Chapter 17: in an isosceles triangle the angles opposite the equal sides are equal, and the three angles add up to 180°.
Writing reasons. In the exam, circle theorem questions say “give reasons for your answer”. You get marks for the angle and for the reason, and the reason must be in the right words. Learn the wording in the box below and use it exactly. Each section of this chapter shows you how a theorem works and why it is true.
The reasons you must be able to write
- Angle in a semicircle is 90°.
- The angle at the centre is twice the angle at the circumference.
- Angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral add up to 180°.
- The angle between a tangent and a radius is 90°.
- Tangents from an external point are equal in length.
- The angle between a tangent and a chord is equal to the angle in the alternate segment.
- The perpendicular from the centre to a chord bisects the chord.
- Also use: base angles of an isosceles triangle are equal; angles in a triangle add up to 180°; angles on a straight line add up to 180°; angles in a quadrilateral add up to 360°.
Watch out
A chord is not the same as a radius. A radius has one end at the centre. A chord has both ends on the circle. The only chord that passes through the centre is the diameter.Method. Match each description to the exact circle word.
- Look at CD. It joins two points on the circle and passes through the centre O.(a) CD is a diameter (a chord through the centre)
- Look at the region OEF. It is bounded by two radii (OE and OF) and an arc.(b) a sector
- Look at the region by AB. AB is a chord that does not pass through the centre. The region between the chord and its arc is cut off from the rest of the circle.(c) a segment
- Look at HJ. It touches the circle at one point, G, and does not cross into the circle.(d) a tangent
Check. A sector has two radii as straight sides. A segment has a chord as its straight side. ✓
Watch out. A sector is bounded by radii; a segment is bounded by a chord. Both have an arc.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. Every line from the centre to the circle is a radius, and all radii of one circle are the same length.OA and OB are both radii, so OA = OB.
- Name the triangle. A triangle with two equal sides is isosceles.Triangle OAB is isosceles, with the two equal sides meeting at O.
- Use the angle sum. The three angles of any triangle add up to 180°. The two base angles (at A and B) are equal because the triangle is isosceles.Angle OAB + angle OBA + angle AOB = 180°
2 × angle OAB + 70° = 180° - Solve for the base angle. Take away the known angle, then share the rest equally between the two base angles.2 × angle OAB = 180° − 70° = 110°
angle OAB = 110° ÷ 2 = 55°
Check. The three angles are 55° + 55° + 70° = 180°, which is a full triangle. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. All radii of a circle are the same length.OA and OB are both radii, so OA = OB and triangle OAB is isosceles.
- Write down the other base angle. The base angles of an isosceles triangle are equal.Angle OBA = angle OAB = 25°
- Use the angle sum. The three angles of a triangle add up to 180°, so subtract the two base angles from 180°.Angle AOB = 180° − 25° − 25° = 130°
Check. 25° + 25° + 130° = 180°. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Method. Show that all three sides of triangle OAB are equal. Then the triangle is equilateral and each angle is 60°.
- List the three sides. OA and OB are radii of the circle. AB is the chord, and the question says it is equal to the radius.OA = OB = radius, and AB = radius
- Say what kind of triangle it is. A triangle with three equal sides is equilateral.Triangle OAB is equilateral.
- Use the angle fact. The three angles of a triangle add up to 180°, and in an equilateral triangle they are all equal.180° ÷ 3 = 60°
- Write the conclusion. End a “show that” question with a clear statement.So angle AOB = 60°. ✓
Check. Another way: the base angles are equal and so is the apex, so 3 × 60° = 180°. ✓
Watch out. In a “show that” question, state each fact and reason; do not just draw a number from nowhere.
Exercise 26A Circle words and isosceles triangles
Non-calculatorKey words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.chord: a straight line that joins two points on the circle
- Match it to the question. AB has both ends on the circle and does not pass through the centre.The answer is chord.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.diameter: a chord that passes through the centre
- Match it to the question. CD joins two points on the circle and passes through O.The answer is diameter.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.radius: a straight line from the centre to a point on the circle
- Match it to the question. OE starts at the centre O and ends on the circle at E.The answer is radius.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.sector: a region between two radii and an arc
- Match it to the question. The region has two radii (OE and OF) as straight sides.The answer is sector.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.segment: a region between a chord and an arc
- Match it to the question. The region is cut off by the chord AB.The answer is segment.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.tangent: a straight line that touches the circle at exactly one point
- Match it to the question. HJ touches the circle at G only.The answer is tangent.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.arc: a part of the circumference
- Match it to the question. The curved line from E to F is part of the circle itself. It is the shorter way round, so it is a minor arc.The answer is arc.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.radius: a straight line from the centre to the circle
- Match it to the question. The line starts at the centre.The answer is radius.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.circumference: the distance all the way round a circle
- Match it to the question. This is the length of the circle itself.The answer is circumference.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.diameter: a chord that passes through the centre
- Match it to the question. The line goes through the centre.The answer is diameter.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.semicircle: half of a circle, cut off by a diameter
- Match it to the question. A diameter cuts the circle into two equal halves.The answer is semicircle.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.tangent: a line that touches the circle at exactly one point
- Match it to the question. The line only touches and never goes inside.The answer is tangent.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Key words. The circumference is the distance all the way round a circle (it is also the name of the curved line itself). A circle theorem is a rule that is always true for angles or lines in a circle.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Circle words have exact meanings. Each word names one thing.segment: the region between a chord and an arc
- Match it to the question. The straight edge of the region is a chord.The answer is segment.
Check. Say the meaning in your own words and see if it fits what the question describes. ✓
Watch out. Several words sound alike. A chord joins two points on the circle; a tangent only touches the circle; a radius starts at the centre.
Method. The diameter is twice the radius, because it is made of two radii end to end.
- Write the fact. A diameter passes through the centre, so it is two radii in a straight line.diameter = 2 × radius
- Calculate. 2 × 9 = 18 cm
Check. 18 ÷ 2 = 9. ✓
Watch out. Radius to diameter: double. Diameter to radius: halve.
Method. The radius is half the diameter.
- Write the fact. The diameter is two radii end to end.radius = diameter ÷ 2
- Calculate. 26 ÷ 2 = 13 cm
Check. 2 × 13 = 26. ✓
Watch out. Radius to diameter: double. Diameter to radius: halve.
Method. The diameter is twice the radius.
- Write the fact. The diameter is two radii end to end.diameter = 2 × radius
- Calculate. Double 6.5: double 6 is 12 and double 0.5 is 1.2 × 6.5 = 13 m
Check. 13 ÷ 2 = 6.5. ✓
Watch out. Radius to diameter: double. Diameter to radius: halve.
Method. Compare Dev's description with the exact meanings of the two words.
- Recall what a chord is. A chord has both ends on the circle.chord: joins two points on the circle
- Recall what Dev described. A line from the centre to the circle starts at the centre, which is not on the circle.That line is a radius.
- Write the correction. Dev's line is a radius, not a chord. A chord has both ends on the circle.
Check. Only the diameter is both a chord and passes through the centre. ✓
Watch out. Learn the difference: a radius starts at the centre; a chord never needs the centre.
Method. Compare the chord with two radii, using the fact that the shortest route between two points is a straight line.
- Write the lengths in terms of the radius. OA and OB are both radii, so each equals r.OA + OB = r + r = 2r = the diameter
- Compare. In any triangle, one side is shorter than the other two added together (the straight route is shortest).AB < OA + OB
- State the conclusion. So AB is shorter than the diameter. ✓
Check. If A, O and B were in a straight line then AB would equal the diameter exactly. ✓
Watch out. The chord and the two radii make a triangle only if O is not on the chord.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. Every line from the centre to the circle is a radius, and all radii of one circle are the same length.OA and OB are both radii, so OA = OB.
- Name the triangle. A triangle with two equal sides is isosceles.Triangle OAB is isosceles, with the two equal sides meeting at O.
- Use the angle sum. The three angles of any triangle add up to 180°. The two base angles (at A and B) are equal because the triangle is isosceles.Angle OAB + angle OBA + angle AOB = 180°
2 × angle OAB + 40° = 180° - Solve for the base angle. Take away the known angle, then share the rest equally between the two base angles.2 × angle OAB = 180° − 40° = 140°
angle OAB = 140° ÷ 2 = 70°
Check. The three angles are 70° + 70° + 40° = 180°, which is a full triangle. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. Every line from the centre to the circle is a radius, and all radii of one circle are the same length.OA and OB are both radii, so OA = OB.
- Name the triangle. A triangle with two equal sides is isosceles.Triangle OAB is isosceles, with the two equal sides meeting at O.
- Use the angle sum. The three angles of any triangle add up to 180°. The two base angles (at A and B) are equal because the triangle is isosceles.Angle OAB + angle OBA + angle AOB = 180°
2 × angle OAB + 66° = 180° - Solve for the base angle. Take away the known angle, then share the rest equally between the two base angles.2 × angle OAB = 180° − 66° = 114°
angle OAB = 114° ÷ 2 = 57°
Check. The three angles are 57° + 57° + 66° = 180°, which is a full triangle. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. Every line from the centre to the circle is a radius, and all radii of one circle are the same length.OA and OB are both radii, so OA = OB.
- Name the triangle. A triangle with two equal sides is isosceles.Triangle OAB is isosceles, with the two equal sides meeting at O.
- Use the angle sum. The three angles of any triangle add up to 180°. The two base angles (at A and B) are equal because the triangle is isosceles.Angle OAB + angle OBA + angle AOB = 180°
2 × angle OAB + 110° = 180° - Solve for the base angle. Take away the known angle, then share the rest equally between the two base angles.2 × angle OAB = 180° − 110° = 70°
angle OAB = 70° ÷ 2 = 35°
Check. The three angles are 35° + 35° + 110° = 180°, which is a full triangle. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. All radii of a circle are the same length.OA and OB are both radii, so OA = OB and triangle OAB is isosceles.
- Write down the other base angle. The base angles of an isosceles triangle are equal.Angle OBA = angle OAB = 48°
- Use the angle sum. The three angles of a triangle add up to 180°, so subtract the two base angles from 180°.Angle AOB = 180° − 48° − 48° = 84°
Check. 48° + 48° + 84° = 180°. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. All radii of a circle are the same length.OA and OB are both radii, so OA = OB and triangle OAB is isosceles.
- Write down the other base angle. The base angles of an isosceles triangle are equal.Angle OBA = angle OAB = 71°
- Use the angle sum. The three angles of a triangle add up to 180°, so subtract the two base angles from 180°.Angle AOB = 180° − 71° − 71° = 38°
Check. 71° + 71° + 38° = 180°. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Key words. A radius joins the centre to the circle. A chord joins two points on the circle. Isosceles means two equal sides, which gives two equal angles.
Method. Two radii of a circle are equal, so any triangle made from two radii and a chord is isosceles. Then use the isosceles and angle-sum facts.
- Spot the radii. All radii of a circle are the same length.OA and OB are both radii, so OA = OB and triangle OAB is isosceles.
- Write down the other base angle. The base angles of an isosceles triangle are equal.Angle OBA = angle OAB = 15°
- Use the angle sum. The three angles of a triangle add up to 180°, so subtract the two base angles from 180°.Angle AOB = 180° − 15° − 15° = 150°
Check. 15° + 15° + 150° = 180°. ✓
Watch out. Do not give the base angles as the angle at the centre. In triangle OAB the equal angles are the two at the circumference, A and B.
Method. Triangle OAB is isosceles (OA = OB are radii), so angle OBA = angle OAB. The three angles add up to 180°.
- Write the three angles. The base angles are both (2x + 5)°.angle OAB = angle OBA = (2x + 5)°, angle AOB = (x + 10)°
- Add them to make 180. The angles of a triangle add up to 180°.(2x + 5) + (2x + 5) + (x + 10) = 180
- Collect like terms. Add the x terms: 2 + 2 + 1 = 5. Add the numbers: 5 + 5 + 10 = 20.5x + 20 = 180
- Subtract 20 from both sides. We want the number terms on the other side.5x = 180 − 20
5x = 160 - Divide both sides by 5. 5x means 5 × x, so dividing by 5 leaves x on its own.x = 160 ÷ 5 = 32
Check. x = 32: the angles are 69°, 69° and 42°, which add up to 180°. ✓
Watch out. Remember there are two equal base angles, so the expression (2x + 5) appears twice.
Method. Substitute x = 32 into the expression for angle OAB.
- Write the expression. angle OAB = (2x + 5)°
- Substitute. 2 × 32 + 5 = 64 + 5 = 69
- Write the answer. angle OAB = 69°
Check. 69° + 69° + 42° = 180°. ✓
Watch out. Give the angle itself when the question asks for the angle, not just x.
Method. Find the angle at the centre of one sector, then use the isosceles triangle made by the two radii.
- Find angle AOB. The 5 sectors are equal and fill the whole turn at the centre, which is 360°.360° ÷ 5 = 72°
- Spot the isosceles triangle. OA and OB are radii, so they are equal and the base angles are equal.angle OAB = angle OBA
- Use the angle sum. The angles of triangle OAB add up to 180°.angle OAB = (180° − 72°) ÷ 2 = 108° ÷ 2 = 54°
Check. 54° + 54° + 72° = 180°. ✓
Watch out. The angle at the centre of a sector comes from dividing 360°, not 180°.
Method. Each pair of radii makes an isosceles triangle. Find the base angles at B in each triangle and add them.
- Spot two isosceles triangles. OA, OB and OC are all radii, so they are all equal. That makes triangles OAB and OBC isosceles.OA = OB = OC (radii)
- Find angle OBA. In triangle OAB the base angles are equal, and the angles add up to 180°.Angle OBA = (180° − 100°) ÷ 2 = 40° (the base angles of an isosceles triangle are equal; the angles in a triangle add up to 180°)
- Find angle OBC. Do the same in triangle OBC.Angle OBC = (180° − 80°) ÷ 2 = 50°
- Add the two parts of angle ABC. The line BO splits angle ABC into angle ABO and angle OBC.Angle ABC = 40° + 50° = 90°
Check. Second route: the arc AC not containing B has centre angle 360° − 100° − 80° = 180°. Half of it (the angle at the circumference) is 90°. ✓
Watch out. Angle ABC is made of two angles. Do not stop after finding only one of them.
Method. Each pair of radii makes an isosceles triangle. Find the base angles at B in each triangle and add them.
- Spot two isosceles triangles. OA, OB and OC are all radii, so they are all equal. That makes triangles OAB and OBC isosceles.OA = OB = OC (radii)
- Find angle OBA. In triangle OAB the base angles are equal, and the angles add up to 180°.Angle OBA = (180° − 130°) ÷ 2 = 25° (the base angles of an isosceles triangle are equal; the angles in a triangle add up to 180°)
- Find angle OBC. Do the same in triangle OBC.Angle OBC = (180° − 70°) ÷ 2 = 55°
- Add the two parts of angle ABC. The line BO splits angle ABC into angle ABO and angle OBC.Angle ABC = 25° + 55° = 80°
Check. Second route: the arc AC not containing B has centre angle 360° − 130° − 70° = 160°. Half of it (the angle at the circumference) is 80°. ✓
Watch out. Angle ABC is made of two angles. Do not stop after finding only one of them.
Method. Use the isosceles triangle made by the two radii, and the angle sum of a triangle.
- Show the triangle is isosceles. OA and OB are both radii, so OA = OB.triangle OAB is isosceles
- Write the other base angle. The base angles of an isosceles triangle are equal.angle OBA = angle OAB = x
- Use the angle sum. The three angles add up to 180°.angle AOB + x + x = 180°
- Rearrange. Take 2x from both sides.angle AOB = 180° − 2x ✓
Check. Try x = 50°: the formula gives 180° − 100° = 80°, and 50 + 50 + 80 = 180. ✓
Watch out. Name the facts you use: radii are equal, base angles are equal, angle sum is 180°.
Method. Use the angle sum 180° and the fact that the base angles are equal.
- Suppose a base angle were 90° or more. We test this by assuming it and looking for a contradiction.base angle ≥ 90°
- Use the second base angle. The base angles of an isosceles triangle are equal, so both are at least 90°.two base angles add to at least 180°
- Compare with the angle sum. The three angles must add up to exactly 180°, but the angle at O is more than 0°.That would make the total more than 180°, which is impossible. So each base angle is less than 90°.
Check. Look at the examples: the base angles were 70°, 57°, 35°, all below 90°. ✓
Watch out. This is why the base angles of triangle OAB are always acute.