Question

8 in △pqr and △stu, ∠p = 50°, ∠q = 60°, and ∠s = 50°, ∠t = 60°, which postulate can be used to prove similarity?

• a sas
• b ssa
• c aa
• d sss

9 which of the following is not a step in proving triangle similarity using aa?

• a show that three angles are congruent
• b identify two pairs of corresponding angles
• c conclude the triangles are similar
• d prove that the triangles are congruent

10 which of the following represents a correct ratio if triangles are similar by sas?

• a ab/de = ac/df
• b ab + de = ac + df
• c ab × de = ac × df
• d ab - de = ac - df

11 which of the following is necessary to prove similarity using sas similarity?

• a one pair of congruent angles and all sides equal
• b one pair of congruent angles and the sides around the angle proportional
• c one pair of proportional sides
• d two pairs of congruent angles

12 which of the following statements is true if △pqr ~ △xyz?

• a ∠q = ∠z
• b ∠p = ∠x
• c pr = xy
• d pq = yz

13 which of the following best describes why triangles △abc and △def are similar by aa similarity?

• a two angles of one triangle are congruent to two angles of the other triangle
• b their areas are equal
• c their corresponding side lengths are equal
• d they have the same perimeter

14 what does the scale factor indicate in similar triangles?

• a how much one triangle is scaled up or down
• b the area ratio
• c the difference in angle measurements
• d the perimeter difference

Explanation:

• Question 8: In \(\triangle PQR\) and \(\triangle STU\), \(\angle P = 50^{\circ}\), \(\angle Q=60^{\circ}\), \(\angle S = 50^{\circ}\), \(\angle T = 60^{\circ}\). By the AA (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), we can prove similarity.
• Question 9: When proving triangle similarity using AA, we do not prove the triangles are congruent. Congruent triangles are a special case of similar triangles with a scale factor of \(1\), but AA similarity just requires two pairs of congruent angles.
• Question 10: For SAS (Side - Angle - Side) similarity, the ratio of the sides around the included angle should be proportional. If \(\triangle ABC\sim\triangle DEF\) by SAS similarity, \(\frac{AB}{DE}=\frac{AC}{DF}\) (assuming \(\angle A=\angle D\)).
• Question 11: The SAS similarity criterion states that if one pair of angles is congruent and the sides around the angle are proportional, then the triangles are similar.
• Question 12: If \(\triangle PQR\sim\triangle XYZ\), then corresponding angles are equal. So \(\angle P=\angle X\), \(\angle Q=\angle Y\), \(\angle R=\angle Z\).
• Question 13: The AA similarity criterion is based on the fact that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
• Question 14: The scale factor in similar triangles is the ratio of corresponding side lengths.

Answer:

• Question 8: C. AA
• Question 9: d. Prove that the triangles are congruent
• Question 10: a. \(AB:DE = AC:DF\)
• Question 11: b. One pair of congruent angles and the sides around the angle proportional
• Question 12: b. \(\angle P=\angle X\)
• Question 13: a. Two angles of one triangle are congruent to two angles of the other triangle
• Question 14: b. The side ratio