Question

1. by definition, a parallelogram is a quadrilateral in which both pairs of opp

use this definition in your proof.
given: ( a b c d ) is a parallelogram
prove: ( overline{a d} cong overline{b c} )

1. for each part below, use the figure to fill in the blank.

if necessary, you may learn what the markings on a figure indicate.
(a) find ( m angle m ).
(b) find ( m angle q ).

Explanation:

Step1: Analyze the triangle in part (a)

The triangle in part (a) is an equilateral triangle (all sides are marked equal). In an equilateral triangle, all angles are equal. The sum of angles in a triangle is \(180^{\circ}\). Let each angle be \(x\). Then \(x + x+ x=180^{\circ}\), so \(3x = 180^{\circ}\), and \(x = 60^{\circ}\). So \(m\angle M=60^{\circ}\).

Step2: Analyze the triangle in part (b)

The triangle in part (b) is an isosceles triangle (\(SR = QR = 10\)). The sum of angles in a triangle is \(180^{\circ}\). Let \(m\angle Q=m\angle S=y\) (base - angles of an isosceles triangle are equal). We know that \(m\angle R = 110^{\circ}\). Then \(y + y+110^{\circ}=180^{\circ}\), \(2y=180^{\circ}- 110^{\circ}=70^{\circ}\), \(y = 35^{\circ}\). So \(m\angle Q = 35^{\circ}\).

Answer:

(a) \(m\angle M = 60^{\circ}\)
(b) \(m\angle Q=35^{\circ}\)