Question

1. which similarity criterion requires two angles of one triangle to be equal to two angles of another triangle?

a. sss
b. hl
c. sas
d. aa

1. prove that △pqr ≅ △xyz using asa congruence where ∠p = ∠x = 40°, ∠q = ∠y = 60°, and pq = xy = 5. fill in the missing congruence criterion.

a. sss congruence criterion
b. asa congruence criterion
c. hl congruence criterion
d. aas congruence criterion
(table: statement, reason: ∠p = ∠x = 40° (given: corresponding angles are congruent), pq = xy = 5 (given: corresponding sides are congruent), ∠q = ∠y = 60° (given: corresponding angles are congruent), △pqr ≅ △xyz (______))

1. a ladder leans against a wall, forming a right triangle. the ladder is 12 ft long and the base is 5 ft from the wall. what is the height of the ladder on the wall?

a. 10 ft
b. 15 ft
c. 12 ft
d. 7 ft

Explanation:

Question 7

• Recall triangle similarity criteria:
• SSS (Side - Side - Side) similarity requires all three corresponding sides to be in proportion.
• HL (Hypotenuse - Leg) is for right - triangle congruence, not similarity in general and doesn't involve two angles.
• SAS (Side - Angle - Side) similarity requires two sides to be in proportion and the included angle to be equal.
• AA (Angle - Angle) similarity states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. So the criterion that requires two angles of one triangle to be equal to two angles of another triangle is AA.

• The ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
• In the given problem, we have $\angle P=\angle X$, $PQ = XY$ (the included side between $\angle P$ and $\angle Q$ and between $\angle X$ and $\angle Y$ respectively), and $\angle Q=\angle Y$. So the congruence criterion used is ASA.

Step1: Identify the triangle type and formula

We have a right triangle where the ladder is the hypotenuse ($c = 12$ ft), the distance from the base of the ladder to the wall is one leg ($a = 5$ ft), and the height on the wall is the other leg ($b$). We use the Pythagorean theorem: $a^{2}+b^{2}=c^{2}$.

Step2: Solve for \(b\)

Rearrange the formula to solve for \(b\): $b=\sqrt{c^{2}-a^{2}}$. Substitute $c = 12$ and $a = 5$ into the formula: $b=\sqrt{12^{2}-5^{2}}=\sqrt{144 - 25}=\sqrt{119}\approx10.9$ ft, which is approximately 11 ft.

Answer:

d. AA

Question 8