Question

determine the value of each variable. then explain, with a test, why the two triangles are similar.
6 △abc is similar to △def.
7 △abc is similar to △def.

Explanation:

Step1: Find angle x (Problem 6)

Sum of angles in triangle is 180°. Missing angle: $180 - 64 - y$. Wait, similar triangles have equal angles. Assume corresponding angles: ∠A=∠F=78°, ∠B=∠D=y=64°, so ∠C=∠E=180-78-64=38°. So x=78°, y=64°.

Step2: Similarity test (Problem6)

AA test: Two pairs of equal angles (78° and 64°), so similar.

Step3: Find b and z (Problem7)

Corresponding sides: AB/EF = BC/FD = AC/ED. Assume ∠B=∠F=35°, so z=35°. Ratio: BC/FD=4.5/3=1.5, so AB=EF×1.5=5.2×1.5=7.8 cm.

Step4: Similarity test (Problem7)

SAS test: Included angle (35°) equal, sides proportional (1.5), so similar.

Answer:

Problem6: x=78°, y=64°; AA similarity.
Problem7: b=7.8 cm, z=35°; SAS similarity.