1. **Problem statement:** Given a triangle with an exterior angle of 108° and one interior angle of 40°, find: a) The other two interior angles. b) The other two exterior angles. 2. **Key formula and rules:** - The exterior angle of a triangle equals the sum of the two opposite interior angles. - The sum of interior angles in any triangle is 180°. - Exterior and interior angles at the same vertex are supplementary (sum to 180°). 3. **Part a: Find the other two interior angles.** - Let the interior angle adjacent to the exterior angle be $x$. - Since exterior and interior angles at the same vertex sum to 180°, $$x + 108 = 180$$ $$x = 180 - 108 = 72$$ - The two opposite interior angles to the exterior angle are 40° and $72°$. - The third interior angle is the one adjacent to the exterior angle, which is $72°$. - Check sum: $$40 + 72 + 68 = 180$$ But we need to find the third angle, call it $z$: $$40 + 72 + z = 180$$ $$z = 180 - 112 = 68$$ - So the interior angles are 40°, 72°, and 68°. 4. **Part b: Find the other two exterior angles.** - Exterior angles at other vertices are supplementary to their interior angles. - For interior angle 40°, exterior angle is: $$180 - 40 = 140$$ - For interior angle 68°, exterior angle is: $$180 - 68 = 112$$ - So the other two exterior angles are 140° and 112°. --- 5. **Problem statement:** In triangle PBC, AQ is parallel to PC. a) Explain why $y = c$. b) Explain why $x = a + y$. c) Use a and b to prove the exterior angle at B of triangle ABC equals the sum of the two opposite interior angles. 6. **Part a:** - Since AQ is parallel to PC and angle $y$ and angle $c$ are alternate interior angles, they are equal. $$y = c$$ 7. **Part b:** - Angle $x$ is the sum of angle $a$ and angle $y$ because they are adjacent angles on a straight line. $$x = a + y$$ 8. **Part c:** - Substitute $y = c$ into $x = a + y$: $$x = a + c$$ - This shows the exterior angle $x$ at vertex B equals the sum of the two opposite interior angles $a$ and $c$. --- 9. **Problem statement:** Given DX parallel to BC, ZD parallel to AB, and BDY is a straight line. a) Explain why angles BAD and ADZ are equal. 10. **Explanation:** - Since DX is parallel to BC and ZD is parallel to AB, angles BAD and ADZ are corresponding angles formed by parallel lines and a transversal. - Corresponding angles are equal. $$\angle BAD = \angle ADZ$$