Question

use rhombus efgh for #6 - 10. 6. m∠egf = __ 7. m∠ehg = 8. fg = 9. eg = 10. mh = __ state the triangle congruence criteria for the pairs of triangles below. if not congruent, state not congruent. 11. 12. 13. 14. 15. 16.

Explanation:

Step1: Recall rhombus properties

In a rhombus, the diagonals are perpendicular bisectors of each other and bisect the angles of the rhombus.

Step2: Find \(m\angle EGF\)

Since the diagonal \(EF\) bisects \(\angle EFG\) and \(\angle EFG = 10^{\circ}\), then \(m\angle EGF=10^{\circ}\).

Step3: Find \(m\angle EMG\)

The diagonals of a rhombus are perpendicular, so \(m\angle EMG = 90^{\circ}\).

Step4: Find \(FG\)

All sides of a rhombus are equal. Given \(EF = 61\), then \(FG=61\).

Step5: Find \(EG\)

The diagonals of a rhombus bisect each other. Let \(EM=x\) and \(MG = 11\), then \(EG=2\times11 = 22\).

Step6: Find \(MH\)

Since the diagonals of a rhombus bisect each other, \(MH=11\).

Step7: Analyze triangle - congruence for question 11

We see two angles and the included side are equal, so it is ASA (Angle - Side - Angle) congruence.

Step8: Analyze triangle - congruence for question 12

We have two right - angled triangles with a pair of equal legs and the right - angles equal. This is SAS (Side - Angle - Side) congruence.

Step9: Analyze triangle - congruence for question 13

We have two sides and the included angle equal, so it is SAS congruence.

Step10: Analyze triangle - congruence for question 14

We have two right - angled triangles with the hypotenuse and a leg equal, so it is HL (Hypotenuse - Leg) congruence.

Step11: Analyze triangle - congruence for question 15

We have two right - angled triangles with a pair of equal legs and the right - angles equal, so it is SAS congruence.

Step12: Analyze triangle - congruence for question 16

We have two angles and a non - included side equal, so it is AAS (Angle - Angle - Side) congruence.

Answer:

1. \(10^{\circ}\)
2. \(90^{\circ}\)
3. \(61\)
4. \(22\)
5. \(11\)
6. ASA
7. SAS
8. SAS
9. HL
10. SAS
11. AAS