Question

1. (l36) from the given statement below, select the definition, property, postulate, or theorem that leads to the statement you are asked to prove.

given: \\( \overline { a b } = \overline { x y } ; \angle b = \angle y ; \overline { c b } = \overline { z y } \\)
prove: \\( \delta a b c = \delta x y z \\)
a. definition of congruent triangles
b. angle-angle-side theorem
c. side-angle-side postulate
d. side-side-side postulate
e. angle-side-angle postulate

1. (l36) from the given statement below, select the definition, property, postulate, or theorem that leads to the statement you are asked to prove.

given: \\( \angle a \cong \angle x ; \overline { b c } \cong \overline { y z } ; \angle c \cong \angle z \\)
prove: \\( \delta a b c \cong \delta x y z \\)
a. definition of congruent triangles
b. angle-angle-side theorem
c. side-angle-side postulate
d. side-side-side postulate
e. angle-side-angle postulate

1. (l31) in \\( \delta l m n \\), the exterior angle adjacent to \\( \angle l \\) has a measure of \\( 5 x + 12 \\). if \\( m \angle m = 3 x - 2 \\) and \\( m \angle n = 50 \\), find the measure of \\( \angle l \\).

a. 56
b. 78
c. 94
d. 123
e. none of the above

1. (l10) \\( \overrightarrow { v f } \\) is between \\( \overrightarrow { v a } \\) and \\( \overrightarrow { v b } \\). if \\( m \angle a v f = 32, m \angle f v b = 2 x + 9 \\), and \\( m \angle a v b = 3 x - 4 \\), then find \\( m \angle a v b \\).

a. 106
b. 97
c. 84
d. 131
e. none of the above

1. (l16) the measure of an angle is 34 greater than its supplement. find the measures of both angles.

a. 73, 107
b. 24, 128
c. 64, 101
d. 45, 97
e. none of the above

Explanation:

Problem 12

We are given two sides ($\overline{AB}\cong\overline{XY}$ and $\overline{CB}\cong\overline{ZY}$) and the included angle ($\angle B\cong\angle Y$). The Side - Angle - Side (SAS) Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

We are given two angles ($\angle A\cong\angle X$ and $\angle C\cong\angle Z$) and a non - included side ($\overline{BC}\cong\overline{YZ}$). The Angle - Angle - Side (AAS) Theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Step1: Use the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So, \(5x + 12=(3x - 2)+50\).

Step2: Solve the equation for \(x\)

$$
LATEXBLOCK0
$$

Step3: Find the measure of \(\angle L\)

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(\angle L=y\). Then \(y+(3x - 2)+50 = 180\). Substitute \(x = 18\) into \(3x-2\): \(3\times18-2=54 - 2=52\). Then \(y+52 + 50=180\), \(y=180-(52 + 50)=78\).

Answer:

C. Side - Angle - Side Postulate

Problem 13