Question

exit ticket
swbat identify and use properties of trapezoids to solve for unknown angle measures & side lengths
directions: complete the following questions by showing all work and annotations. keep work organized and box any final answer. all work must be shown in order to receive full credit.
#1.) trap is an isosceles trapezoid. circle all true statement(s)
a. \\(\overline{ra} \cong \overline{tp}\\)
b. \\(\overline{rp} \cong \overline{at}\\)
c. \\(\angle art \cong \angle rap\\)
d. \\(\overline{wr} \cong \overline{pw}\\)
(image of trapezoid trap with diagonals intersecting at w)
**#2.) trap is an isosceles trapezoid. \\(m\angle r = (3x)\degree\\), \\(m\angle t = (4x + 96)\degree\\) and \\(m\angle p = (12y)\degree\\). determine the sum of x and y below.
(image of trapezoid trap)

Explanation:

Question #1

• Option a: In an isosceles trapezoid, the non - parallel sides (legs) are congruent, but \( \overline{RA} \) and \( \overline{TP} \) are the bases (parallel sides) and not necessarily congruent in length (they are parallel and of different lengths in a trapezoid), so this is false.
• Option b: In an isosceles trapezoid \( TRAP \), the diagonals are congruent. So \( \overline{RP}\cong\overline{AT} \), this is true.
• Option c: \( \angle ART \) and \( \angle RAP \) are alternate interior angles because \( RA\parallel TP \) (bases of trapezoid) and \( RT \) is a transversal? No, actually in isosceles trapezoid, base angles are equal. But \( \angle ART \) and \( \angle RAP \): Let's consider the sides. \( RA \) is parallel to \( TP \), and \( AR \) is a side. The triangles \( ART \) and \( RAP \): Wait, maybe a better way. In isosceles trapezoid, \( \overline{RT}\cong\overline{AP} \) (legs), \( \overline{RA}\parallel\overline{TP} \), and \( \overline{RP}\cong\overline{AT} \) (diagonals). The triangles \( RAT \) and \( A RP \)? No, actually, \( \triangle RAT\cong\triangle APR \) (by SSS, since \( RT = AP \), \( RA=RA \), \( AT = RP \))? Wait, no, maybe \( \angle ART\cong\angle RAP \) because \( RA\parallel TP \) and \( AR \) is a transversal? No, \( RA \) is a base. Wait, if we look at the sides \( RT \) and \( AP \) (legs) are congruent, \( RA \) is common, and \( AT = RP \) (diagonals). So by SSS, \( \triangle RAT\cong\triangle APR \), so \( \angle ART\cong\angle RAP \), this is true.
• Option d: In an isosceles trapezoid, the diagonals bisect each other? No, the diagonals of an isosceles trapezoid are congruent but do not bisect each other (that's a property of parallelograms). So \( \overline{WR}

ot\cong\overline{PW} \) in general, this is false.

Step 1: Use property of isosceles trapezoid (consecutive angles between bases are supplementary)

In an isosceles trapezoid \( TRAP \), \( RA\parallel TP \), so \( \angle R \) and \( \angle T \) are consecutive angles between the bases and are supplementary. So \( m\angle R + m\angle T=180^{\circ} \)
Given \( m\angle R=(3x)^{\circ} \) and \( m\angle T=(4x + 96)^{\circ} \)
So \( 3x+(4x + 96)=180 \)

Step 2: Solve for \( x \)

Combine like terms: \( 3x+4x+96 = 180 \)
\( 7x+96=180 \)
Subtract 96 from both sides: \( 7x=180 - 96=84 \)
Divide both sides by 7: \( x = \frac{84}{7}=12 \)

Step 3: Use property of isosceles trapezoid (base angles are equal)

In isosceles trapezoid \( TRAP \), \( \angle R \) and \( \angle P \) are base angles (since \( RA\parallel TP \) and \( TR, AP \) are legs), so \( m\angle R=m\angle P \)
We know \( m\angle R=(3x)^{\circ} \) and \( x = 12 \), so \( m\angle R=3\times12 = 36^{\circ} \)
And \( m\angle P=(12y)^{\circ} \), so \( 12y=36 \)

Step 4: Solve for \( y \)

Divide both sides by 12: \( y=\frac{36}{12} = 3 \)

Step 5: Find the sum of \( x \) and \( y \)

\( x + y=12 + 3=15 \)

Answer:

b. \( \overline{RP} \cong \overline{AT} \), c. \( \angle ART \cong \angle RAP \)

Question #2