A)

Question

Use the image to answer the question.

Quadrilateral upper Q upper U upper A upper D is drawn within a circle such that all vertices touch the edge of the circle.

Apply properties of quadrilaterals inscribed in a circle to determine the mAD
if mQU=55°
and mAU=112°
, while the m∠UAD=100°
.

(1 point)
mAD=
$$°
Question 2
A)Quadrilateral ABCD
is inscribed in circle O
. The m∠B=4y+27
and m∠D=y−12
. Determine the m∠D
.(1 point)
m∠D=
$$°
Question 3
A)A quadrilateral is inscribed in a circle. Two consecutive angles measure 88° and 108°. The angle opposite the 108° angle is labeled x°
. Find the value of x.(1 point)
x=
$$°
Question 4
A)A quadrilateral is inscribed in a circle. Two consecutive angles measure 88° and 108°. The angle opposite the 88° angle is labeled y°
. Find the value of y.(1 point)
y=
$$°
Question 5
A)
Use the image to answer the question.

Quadrilateral upper M upper A upper T upper H is drawn within a circle such that all vertices touch the edge of the circle. The arc joining upper A and upper T is labeled 50 degrees. The arc joining upper T and upper H is marked 170 degrees.

The image shows an inscribed quadrilateral. The mAM=54°
. Determine the m∠A
.

(1 point)
m∠A=
$$°

Answer 1

To solve the questions related to the properties of quadrilaterals inscribed in a circle, we can apply key properties such as the fact that the opposite angles of a quadrilateral that is inscribed in a circle sum to 180°, as well as relationships between angles and arcs.

Question 1

Given:

• m∠QUAD = 55°
• m∠AUAD = 112°
• m∠UAD = 100°

To find m∠AD, we use the property that the opposite angles of a cyclic quadrilateral sum to 180°:

m∠QUAD + m∠AD = 180° m∠AD = 180° - m∠QUAD m∠AD = 180° - 55° = 125°

Therefore, m∠AD = 125°.

Question 2

We know:

• m∠B = 4y + 27
• m∠D = y - 12

Since ABCD is cyclic, opposite angles sum to 180°:

m∠B + m∠D = 180° 4y + 27 + (y - 12) = 180 5y + 15 = 180 5y = 165 y = 33

Now substitute y back into m∠D:

m∠D = y - 12 = 33 - 12 = 21°

Therefore, m∠D = 21°.

Question 3

Given the two consecutive angles of a cyclic quadrilateral:

• First angle = 88°
• Second angle = 108°

Let the angle opposite the 108° angle be labeled as x°.

Using the property that opposite angles sum to 180°:

x + 108° = 180° x = 180° - 108° x = 72°

Thus, x = 72°.

Question 4

Similar to question 3:

Given:

• First angle = 108°
• Second angle = 88° Let the angle opposite 88° be labeled y°.

Using the property that opposite angles sum to 180°:

y + 88° = 180° y = 180° - 88° y = 92°

Therefore, y = 92°.

Question 5

We are given:

• Arc AT = 50°
• Arc TH = 170°
• m∠AMTH = 54°

To find m∠A, we use the relationship between the angle formed by two chords and the measure of the arc they intercept. The measure of angle A will be half the measure of the arc that it intercepts.

The arc intercepted by angle A (arc AT) is the arc opposite to angle MATH, which consists of the total arc minus the arc MA and arc TH:

The arc opposite M (the angle we know) would be arc TH + arc AT:

Arc opposite A = 50° + 170° = 220°.

Now to find m∠A:

m∠A = (arc opposite A) / 2 = (220°) / 2 = 110°.

Thus, m∠A = 110°.