Question 1

Question

A)
Use the image to answer the question.

A rhombus labeled upper A upper B upper C upper D has 2 interior, diagonal lines.

In the diagram of the rhombus ABCD, m∠C=110°. Use the CPCTC Theorem to prove that the diagonals BD¯¯¯¯¯¯¯¯ and AC¯¯¯¯¯¯¯¯ are perpendicular by finding the m∠D in △DEC.

(1 point)
Responses

55°
55°

35°
35°

125°
125°

70°
70°
Question 2
A)Rhombus QRST has diagonal lines QS¯¯¯¯¯¯¯ and RT¯¯¯¯¯¯¯ that intersect at U. If QS¯¯¯¯¯¯¯=30 and RT¯¯¯¯¯¯¯=16, what is the length of one of the sides of the rhombus?(1 point)
Responses

14
14

46
46

17
17

34
34
Question 3
A)
Use the image to answer the question.

A parallelogram's top left vertex is given as upper A, its top right vertex is given as upper B, its bottom left vertex is given as upper D, and its bottom right vertex is given as upper C. A diagonal line connects upper B and upper D.

In the diagram, AD¯¯¯¯¯¯¯¯=13 and DB¯¯¯¯¯¯¯¯=24. Use the CPCTC Theorem to prove that the diagonal of the rhombus bisects opposite angles by finding the length of DC¯¯¯¯¯¯¯¯. What is the length of DC¯¯¯¯¯¯¯¯?

(1 point)
Responses

5
5

24
24

12
12

13
13
Question 4
A)Rhombus LMNO has a diagonal line MO¯¯¯¯¯¯¯¯¯, with m∠M=74°. What is m∠LOM?(1 point)
Responses

37°
37°

106°
106°

74°
74°

53°
53°
Question 5
A)
Use the image to answer the question.

A parallelogram's top left vertex is given as upper A, its top right vertex is given as upper B, its bottom left vertex is given as upper D, and its bottom right vertex is given as upper C. A diagonal line connects upper B and upper D.

In the diagram, rhombus ABCD has a diagonal line DB¯¯¯¯¯¯¯¯, and m∠A=114°. What is m∠B of rhombus ABCD?

(1 point)
Responses

33°
33°

114°
114°

66°
correct math answers only)

Answer 1

Here are the correct mathematical answers to each of the questions:

Question 1: To find m∠D in triangle DEC given that m∠C = 110°, we note that in triangle ABC (where ABCD is a rhombus), angles A and C must be equal because opposite angles of a rhombus are equal. Therefore, m∠A = 110° as well, and since the sum of angles in a triangle is 180°, we have: m∠D = 180° - m∠A - m∠C = 180° - 110° - 110° = -40°. This doesn't make sense, so let's consider triangle DEC again: Since in a rhombus the diagonals bisect each angle, m∠D is half of 110°, so m∠D = 110°/2 = 55°.

• Answer: 55°

Question 2: In a rhombus, the diagonals bisect each other at right angles. The lengths of the halves of the diagonals (at their intersection U) can form a right triangle. Each half of QS is 15, and each half of RT is 8. Using the Pythagorean theorem: Side length = √(15² + 8²) = √(225 + 64) = √289 = 17.

• Answer: 17

Question 3: The lengths of the diagonals in a parallelogram or rhombus bisect each other. Since AD = 13 and DB = 24, we can find DC. Since it forms two triangles (ABD and DBC), both triangles are congruent, thus DC must also be equivalent to AD.

• Answer: 13

Question 4: In a rhombus, the opposite angles are equal and adjacent angles are supplementary. Therefore, m∠LON = 106°, which means that if m∠M = 74° then m∠LOM would be half of the other angle adjacent in the triangle. Since m∠L should be 180° - m∠M = 180° - 74° = 106°. m∠LOM = m∠M / 2 = 74° / 2 = 37°.

• Answer: 37°

Question 5: In a rhombus, opposite angles are equal and adjacent angles are supplementary. Therefore, if m∠A = 114°, then m∠B = 180° - m∠A = 180° - 114° = 66°.

• Answer: 66°

Each answer corresponds to the properties of rhombuses and parallelograms.