Question

practice 8.3: trapezoids and kites
name: __________ period: __
pqrs is an isosceles trapezoid.

1. label the bases and legs.

name two pairs of congruent angles: $angle p cong angle q$ and $angle s cong angle r$
name a pair of congruent segments: $overline{ps} cong overline{qr}$

abcd is an isosceles trapezoid. $overline{xy}$ is a midsegment.

1. $ax = 4$

find $cd$.

1. $mangle abc = 110^circ$

find $mangle bad$.

1. $mangle bad = 65^circ$

find $mangle cda$.

Explanation:

Step1: Solve for CD (midsegment rule)

In an isosceles trapezoid, the midsegment length equals $\frac{1}{2} \times (\text{base}_1 + \text{base}_2)$. Here, $AX$ is half of base $AB$, so $AB = 2 \times AX = 2 \times 4 = 8$. Let $XY$ be the midsegment, but since $XY \parallel AB \parallel CD$ and $AX = XB$, $DY = YC$, so $CD = AB = 8$.

Step2: Solve for $m\angle BAD$ (consecutive angles)

In a trapezoid, consecutive angles between bases are supplementary. $m\angle ABC + m\angle BAD = 180^\circ$
$m\angle BAD = 180^\circ - 110^\circ = 70^\circ$

Step3: Solve for $m\angle CDA$ (isosceles trapezoid angles)

In an isosceles trapezoid, base angles are congruent. $\angle BAD \cong \angle CDA$, so $m\angle CDA = 65^\circ$

Answer:

1. $CD = 8$
2. $m\angle BAD = 70^\circ$
3. $m\angle CDA = 65^\circ$