Question

answer the following questions about this figure. part a if m∠3 = 135°, what is the m∠1? part b derrick says that △def is an isosceles triangle. is derrick correct? explain.

Explanation:

Step1: Find m∠2

Since ∠2 and ∠3 are a linear - pair (sum of angles in a linear pair is 180°), we have m∠2=180° - m∠3. Given m∠3 = 135°, then m∠2=180°−135° = 45°.

Step2: Find m∠1

In right - triangle DEF with ∠D = 90°, using the angle - sum property of a triangle (the sum of interior angles of a triangle is 180°), we know that m∠1+ m∠2+ m∠D=180°. Substituting m∠D = 90° and m∠2 = 45° into the equation, we get m∠1+45°+90° = 180°. Then m∠1=180°−90°−45° = 45°.

Step3: Check if △DEF is isosceles

In △DEF, we found that m∠1 = 45° and m∠2 = 45°. Since two angles of the triangle (∠1 and ∠2) are equal, the sides opposite these equal angles are also equal. By the definition of an isosceles triangle (a triangle with at least two equal - length sides), △DEF is an isosceles triangle.

Answer:

Part A: m∠1 = 45°
Part B: Derrick is correct. Since m∠1 = m∠2 = 45°, △DEF has two equal angles and thus two equal sides, making it an isosceles triangle.