Question

1. if two triangles are similar by sas, what can be said about their corresponding angles? a. they are complementary b. they are congruent c. they are all different d. they are proportional 2. in △ghi and △jkl, if ∠g = ∠j, gh = 10, gi = 15, and jk = 5, what is the missing side jl? a. 10 cm b. 8.5 cm c. 7 cm d. 7.5 cm 3. which mathematical property justifies why two triangles are similar when two angles are congruent? a. angle sum property of a triangle b. transitive property c. reflexive property d. distributive property

Explanation:

Step1: Recall SAS similarity property

If two triangles are similar by SAS (Side - Angle - Side) similarity criterion, then their corresponding angles are congruent.

Step2: Solve for the missing side in similar triangles

Since \(\triangle GHI\sim\triangle JKI\) (by the given equal angle and proportional sides), we use the proportion \(\frac{GH}{JK}=\frac{GI}{JI}\). Substituting \(GH = 10\), \(GI=15\), and \(JK = 5\) into the proportion \(\frac{10}{5}=\frac{15}{JI}\), we cross - multiply to get \(10\times JI=5\times15\), then \(JI=\frac{5\times15}{10}=7.5\).

Step3: Recall similarity criterion for two angles

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar by the AA (Angle - Angle) similarity criterion. This is based on the Angle - Sum Property of a Triangle. Since the sum of angles in a triangle is \(180^{\circ}\), if two pairs of angles are equal, the third pair must also be equal.

Answer:

1. B. They are congruent
2. d. \(7.5\ cm\)
3. a. Angle Sum Property of a Triangle