Mathematics • Year 8 • Unit 3 • Lesson 15
Angles in Triangles
Build fluency with the angle sum (always 180°), the exterior angle theorem (= sum of two remote interior angles), and triangle classification (acute, right, obtuse; equilateral, isosceles, scalene).
1. I do, fully worked example
Two big rules: (1) interior angles add to 180°; (2) an exterior angle = sum of the two REMOTE interior angles.
Problem. A triangle has angles 55°, 72°, and x°. Find x.
Step 1, Write the angle-sum equation.
55 + 72 + x = 180
Reason: the sum of the interior angles of ANY triangle is 180°. Always.
Step 2, Simplify.
127 + x = 180
Step 3, Solve.
x = 180 − 127 = 53°
Step 4, Classify the triangle.
All three angles (55°, 72°, 53°) are less than 90° → ACUTE. All three are different → SCALENE.
Answer: x = 53°; the triangle is acute and scalene.
2. We do, fill in the missing steps
An isosceles triangle has a vertex angle of 80°. Find the two equal base angles. 4 marks
Step 1, Write down what we know about isosceles triangles:
An isosceles triangle has ____ equal sides and ____ equal base angles.
Step 2, Set up the angle-sum equation (let each base angle = b):
80 + b + b = ______ ⟶ 80 + 2b = ______
Step 3, Solve for b:
2b = ______ − 80 = ______ ⟶ b = ______ ÷ 2 = ______°
Step 4, State the answer:
Each base angle = ______°
3. You do, independent practice
Show all working AND a brief reason. Foundation: angle sum. Standard: exterior and isosceles. Extension: algebra.
Foundation, find the missing angle
3.1 Triangle angles: 60°, 80°, x°. Find x. 1 mark
3.2 A triangle has two angles equal to 45° each. Find the third angle and classify the triangle. 1 mark
3.3 A right-angled triangle has one acute angle of 35°. Find the other acute angle. 1 mark
3.4 A triangle has angles 95°, 32°, and y°. Find y and classify the triangle. 2 marks
Standard, exterior angle and isosceles
3.5 An exterior angle of a triangle is 130°. One remote interior angle is 70°. Find the other remote interior angle. 2 marks
3.6 An isosceles triangle has a vertex angle of 50°. Find the two equal base angles. 2 marks
Extension, algebraic triangles
3.7 A triangle has angles x, 2x, and 3x. Find all three angles and classify the triangle. 2 marks
3.8 A triangle has angles (2x + 10)°, (x + 20)°, and x°. Find x, all angles, and classify by angle type. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (isosceles, vertex 80°)
Step 1: 2 equal sides and 2 equal base angles.
Step 2: 80 + 2b = 180.
Step 3: 2b = 180 − 80 = 100; b = 100 ÷ 2 = 50°.
Step 4: Each base angle = 50°.
3.1, Angles 60°, 80°, x°
x = 180° − 60° − 80° = 40°.
3.2, Two angles = 45° each
Third angle = 180° − 45° − 45° = 90°. This is a right-angled isosceles triangle.
3.3, Right triangle, one acute = 35°
Other acute angle = 180° − 90° − 35° = 55°. (Or: the two acute angles in a right triangle add to 90°, so other = 90° − 35° = 55°.)
3.4, Angles 95°, 32°, y°
y = 180° − 95° − 32° = 53°. One angle (95°) is greater than 90°, so this is an obtuse scalene triangle.
3.5, Exterior 130°, one remote = 70°
Exterior angle = sum of two remote interior angles: 130° = 70° + other → other = 60°.
3.6, Isosceles, vertex 50°
Each base angle = (180° − 50°) / 2 = 130° / 2 = 65°.
3.7, Angles x, 2x, 3x
x + 2x + 3x = 180 → 6x = 180 → x = 30°. Angles: 30°, 60°, 90°. One angle is 90°, all different → right-angled scalene triangle.
3.8, Angles (2x+10), (x+20), x
(2x + 10) + (x + 20) + x = 180 → 4x + 30 = 180 → 4x = 150 → x = 37.5°. Angles: 2(37.5)+10 = 85°, 37.5+20 = 57.5°, 37.5°. All less than 90° → acute scalene.