Question

1. in the accompanying diagram (l_1parallel l_2) are cut by a transversal (t). (angle4) and (angle5) can be classified as (1) vertical angles (2) same - side interior angles (3) corresponding angles (4) alternate interior angles 3. in the diagram below, (mangle tsv = 21^{circ}) and (mangle rsv=48^{circ}). determine (mangle rst). (1) (27^{circ}) (2) (42^{circ}) (3) (69^{circ}) (4) (132^{circ}) 4. given isosceles triangle (abc), (mangle b = 20^{circ}) and (overline{ab}congoverline{ac}). which of the following choices are true? (1) (mangle a = 20^{circ}) and (mangle c = 140^{circ}) (2) (mangle a = 140^{circ}) and (mangle c = 20^{circ}) (3) (mangle a = 20^{circ}) and (mangle c = 50^{circ}) (4) (mangle a = 50^{circ}) and (mangle c = 20^{circ})

Explanation:

2.

Step1: Recall angle - pair relationships

When two parallel lines \(l_1\) and \(l_2\) are cut by a transversal \(T\), vertical angles are opposite each other at the intersection of two lines, same - side interior angles are on the same side of the transversal and inside the two parallel lines, corresponding angles are in the same relative position with respect to the parallel lines and the transversal, and alternate interior angles are between the two parallel lines and on opposite sides of the transversal. \(\angle4\) and \(\angle5\) are alternate interior angles.

Step1: Use the angle - addition postulate

The angle - addition postulate states that if point \(V\) is in the interior of \(\angle RST\), then \(m\angle RST=m\angle RSV + m\angle TSV\).

Step2: Substitute the given angle measures

We know that \(m\angle TSV = 21^{\circ}\) and \(m\angle RSV=48^{\circ}\). So \(m\angle RST=21^{\circ}+48^{\circ}=69^{\circ}\).

Step1: Recall the properties of an isosceles triangle

In an isosceles triangle \(ABC\) with \(\overline{AB}\cong\overline{AC}\), \(\angle B=\angle C\). Given \(m\angle B = 20^{\circ}\), then \(m\angle C = 20^{\circ}\).

Step2: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle A=x\). Then \(x + 20^{\circ}+20^{\circ}=180^{\circ}\), so \(x=180^{\circ}-(20^{\circ}+20^{\circ}) = 140^{\circ}\).

Answer:

(4) Alternate interior angles

3.