Question

for the parallelogram, if $m\angle2 = 5x - 28$ and $m\angle4 = 3x - 10$, find $m\angle3$. (the diagram is not to scale.)
options: 17 (a), 173, 9, 163
9.
a \underline{\quad\quad} is a parallelogram with four congruent sides and four right angles. (square, kite, rhombus)
10.
a \underline{\quad\quad} is a parallelogram with four congruent sides. (kite, square, rhombus)

Explanation:

Question 1 (Angle in Parallelogram)

Step1: Recall Parallelogram Angle Property

In a parallelogram, opposite angles are equal. So, \( m\angle 2 = m\angle 4 \) (wait, no—wait, in a parallelogram, consecutive angles are supplementary, and opposite angles are equal. Wait, looking at the diagram, \( \angle 2 \) and \( \angle 4 \): wait, maybe \( \angle 2 \) and \( \angle 4 \) are equal? Wait, no, maybe I misread. Wait, in a parallelogram, opposite angles are equal. So if \( \angle 2 \) and \( \angle 4 \) are opposite? Wait, the diagram: let's assume that \( \angle 2 \) and \( \angle 4 \) are equal (opposite angles). So set \( 5x - 28 = 3x - 10 \)? Wait, no, that would give \( 5x - 3x = 28 - 10 \), \( 2x = 18 \), \( x = 9 \). Then \( m\angle 2 = 5(9) - 28 = 45 - 28 = 17 \), \( m\angle 4 = 3(9) - 10 = 27 - 10 = 17 \). Then \( \angle 3 \) and \( \angle 2 \) are consecutive angles, so they are supplementary. So \( m\angle 3 = 180 - 17 = 163 \)? Wait, but let's check again. Wait, maybe \( \angle 2 \) and \( \angle 4 \) are equal (opposite angles). So solving \( 5x - 28 = 3x - 10 \): \( 5x - 3x = 28 - 10 \), \( 2x = 18 \), \( x = 9 \). Then \( m\angle 2 = 5(9) - 28 = 17 \), so \( m\angle 3 = 180 - 17 = 163 \).

Step2: Calculate \( m\angle 3 \)

Since consecutive angles in a parallelogram are supplementary, \( m\angle 3 + m\angle 2 = 180^\circ \). We found \( m\angle 2 = 17^\circ \), so \( m\angle 3 = 180 - 17 = 163^\circ \).

A square is a parallelogram (has opposite sides parallel) with four congruent sides (all sides equal) and four right angles (each angle \( 90^\circ \)). A kite is not a parallelogram (only one pair of opposite sides parallel? No, kite has two pairs of adjacent sides equal, not opposite sides parallel). A rhombus has four congruent sides but not necessarily right angles. So the answer is square.

A rhombus is a parallelogram (opposite sides parallel) with four congruent sides (all sides equal). A kite is not a parallelogram. A square is a rhombus with right angles, so a rhombus is the general term for a parallelogram with four congruent sides (square is a special case). So the answer is rhombus.

Answer:

163 (Option D)

Question 9 (Parallelogram with 4 Congruent Sides and 4 Right Angles)