Question

homework
5.04: swbat identify and use properties of trapezoids to solve for unknown angle measures & side lengths
show all work and annotations for full credit. use your quadrilateral property reference sheet!
**#1.) in isosceles trapezoid trap, ( mangle t = 6x - 10 ), ( mangle r = x + 90 ). determine the number of degrees in all four angles.

b.) in isosceles trapezoid trap, ( tp = (7y - 7) ) ft, ( tr = (7y + 7) ) ft, ( ra = (2y + 28) ) ft, and ( ap = (8y + 7) ) ft. determine the perimeter of trap.

**#2.) in trapezoid trap above ( mangle t = (4x)^circ ) and ( mangle r = (8x - 12)^circ ). determine the ( mangle r ) and ( mangle t ).

Explanation:

Part 1a: Angles in Isosceles Trapezoid TRAP

Step1: Recall trapezoid angle property

In an isosceles trapezoid, consecutive angles between the bases are supplementary? No, wait, in a trapezoid (including isosceles), consecutive angles along a leg are supplementary. Wait, actually, in an isosceles trapezoid, base angles are equal, and consecutive angles between the legs are supplementary? Wait, no, let's correct: In a trapezoid, \( TRAP \) – let's assume \( TR \) and \( AP \) are the legs, and \( TA \) and \( RP \) are the bases? Wait, the diagram: points \( R, A, P, T \) – so \( RA \) and \( TP \) are the legs? Wait, no, the first diagram: \( R \) connected to \( A \), \( A \) to \( P \), \( P \) to \( T \), \( T \) to \( R \). So \( RA \) and \( TP \) are the legs, \( RT \) and \( AP \) are the bases? Wait, no, in a trapezoid, there are two parallel sides (bases). So in isosceles trapezoid, the legs are equal, and base angles are equal, and consecutive angles between the bases are supplementary. Wait, actually, in any trapezoid, consecutive angles along a leg are supplementary (since the bases are parallel, so same - side interior angles are supplementary). So \( \angle T \) and \( \angle R \) are consecutive angles along leg \( TR \), so they should be supplementary: \( m\angle T + m\angle R=180^{\circ} \)

Given \( m\angle T = 6x - 10 \), \( m\angle R=x + 90 \)

So \( (6x - 10)+(x + 90)=180 \)

Step2: Solve for x

Combine like terms: \( 6x+x-10 + 90=180 \)

\( 7x+80 = 180 \)

Subtract 80 from both sides: \( 7x=180 - 80=100 \)? Wait, no, 180 - 80 is 100? Wait, 180-80 = 100? Then \( 7x = 100 \)? But 100 is not divisible by 7. Wait, maybe I got the angles wrong. Wait, in an isosceles trapezoid, base angles are equal. Let's re - examine the diagram. The first diagram: \( R \) and \( A \) are top, \( T \) and \( P \) are bottom? Wait, maybe \( \angle T \) and \( \angle P \) are base angles (equal), \( \angle R \) and \( \angle A \) are base angles (equal), and \( \angle T \) and \( \angle R \) are supplementary. Wait, let's check the problem again: "In isosceles trapezoid TRAP, \( m\angle T = 6x - 10 \), \( m\angle R=x + 90 \). Determine the number of degrees in all four angles."

Wait, maybe \( \angle T=\angle P \) and \( \angle R=\angle A \), and \( \angle T+\angle R = 180^{\circ} \) (since they are adjacent angles along the leg). So:

\( 6x-10+x + 90=180 \)

\( 7x + 80=180 \)

\( 7x=100 \)? That can't be. Wait, maybe a typo? Wait, no, maybe I misread the angles. Wait, maybe \( \angle T \) and \( \angle R \) are base angles? No, in isosceles trapezoid, base angles are equal. So if \( \angle T \) and \( \angle R \) are base angles, then \( 6x - 10=x + 90 \)

Let's try that: \( 6x-x=90 + 10 \)

\( 5x=100 \)

\( x = 20 \)

Ah, that makes sense. So maybe the bases are \( TP \) and \( RA \), so \( \angle T \) and \( \angle R \) are base angles? Wait, no, in an isosceles trapezoid, the base angles are the angles adjacent to each base. So if the two bases are \( TR \) and \( AP \), then \( \angle T \) and \( \angle P \) are adjacent to base \( TP \), \( \angle R \) and \( \angle A \) are adjacent to base \( RA \). And \( \angle T=\angle P \), \( \angle R=\angle A \), and \( \angle T+\angle R = 180^{\circ} \). Wait, but if we assume that \( \angle T=\angle R \) (base angles), then:

\( 6x-10=x + 90 \)

\( 5x=100 \)

\( x = 20 \)

Then \( m\angle T=6(20)-10 = 120 - 10=110^{\circ} \)

\( m\angle R=20 + 90 = 110^{\circ} \)

Then \( \angle T \) and \( \angle R \) are equal, and since consecutive angles between the bases are supplementary, \( \angle T+\angle P = 180^{\circ} \), so \( \a…

Step1: Recall isosceles trapezoid side property

In an isosceles trapezoid, the legs are equal. So \( TP = RA \) (legs)

Given \( TP=(7y - 7)\) ft, \( RA=(2y + 28)\) ft

So \( 7y-7=2y + 28 \)

Step2: Solve for y

Subtract \( 2y \) from both sides: \( 7y-2y-7=28 \)

\( 5y-7 = 28 \)

Add 7 to both sides: \( 5y=28 + 7=35 \)

\( y = 7 \)

Step3: Find the lengths of all sides

• \( TP=7y - 7=7(7)-7=49 - 7 = 42 \) ft
• \( RA=2y + 28=2(7)+28=14 + 28 = 42 \) ft (checks out, legs are equal)
• \( TR=7y + 7=7(7)+7=49 + 7 = 56 \) ft
• \( AP=8y + 7=8(7)+7=56 + 7 = 63 \) ft

Step4: Calculate the perimeter

Perimeter \( P=TR + RA+AP + TP \)

\( P = 56+42 + 63+42 \)

\( P=56+(42 + 42)+63 \)

\( P=56 + 84+63 \)

\( P=203 \) ft

Part 2: Angles in Trapezoid TRAP

Step1: Recall trapezoid angle property

In trapezoid \( TRAP \) (assuming \( TR\parallel AP \)), consecutive angles along a leg are supplementary. So \( m\angle T + m\angle R=180^{\circ} \)

Given \( m\angle T=(4x)^{\circ} \), \( m\angle R=(8x - 12)^{\circ} \)

So \( 4x+(8x - 12)=180 \)

Step2: Solve for x

Combine like terms: \( 4x+8x-12 = 180 \)

\( 12x-12 = 180 \)

Add 12 to both sides: \( 12x=180 + 12=192 \)

\( x=\frac{192}{12}=16 \)

Step3: Find the measures of the angles

• \( m\angle T=4x=4\times16 = 64^{\circ} \)
• \( m\angle R=8x - 12=8\times16-12=128 - 12 = 116^{\circ} \)

Answer:

s:

Part 1a:

\( m\angle T = 110^{\circ} \), \( m\angle R = 110^{\circ} \), \( m\angle P = 70^{\circ} \), \( m\angle A = 70^{\circ} \) (assuming the correction where \( \angle T=\angle R \) as base angles, since the other approach gave a non - integer x)

Part 1b:

The perimeter of \( TRAP \) is \(\boldsymbol{203}\) feet.

Part 2:

\( m\angle T = 64^{\circ} \), \( m\angle R = 116^{\circ} \)