Answer:
The true statements:
- AB = 8
- x is seven units greater than y
- m∠ABC = 122°
Step-by-step explanation:
* One case of congruence is SAS that means
- If two sides and including angle between them in the first
triangle equal to the corresponding sides and angle in the
second triangle, then the two triangles are congruent
* In the problem
- Triangle ABC is congruent to triangle PQR using SAS case
* That means
- AB = PQ
- BC = QR
- m∠ABC = m∠PQR
- Where angle ABC including angle between AB and BC
and angle PQR is the including angle between PQ and QR
* Now we will use the information about the lengths of AB
and PQ to find y, and use the measures of angles ABC and
PQR to find x and then chose the correct answer
∵ m∠ABC = 12x + 2
∵ m∠PQR = 142 - 2x
∵ m∠ABC = m∠PQR
∴ 12x + 2 = 142 - 2x ⇒ collect the like terms
∴ 12x + 2x = 142 - 2
∴ 14 x = 140 ⇒ divide both sides by 14
∴ x = 10
* Lets substitute this value of x to find the m∠ABC and m∠PQR
- m∠ABC = 12(10) + 2 = 120 + 2 = 122°
- m∠PQR = 142 - 2(10) = 142 - 20 = 122°
∵ AB = 5y - 7
∵ PQ = 3y - 1
∵ AB = PQ
∴ 5y - 7 = 3y - 1 ⇒ collect the like terms
∴ 5y - 3y = -1 + 7
∴ 2y = 6 ⇒ divide both sides by 2
∴ y = 3
* Lets substitute this value of y to find the AB and PQ
- AB = 5(3) - 7 = 8 units
- PQ = 3(3) - 1 = 8 units
* Lets check the answer to chose the right answer
- 1st answer ⇒ AB = 8 (√)
- 2nd answer ⇒ m∠PQR = 140° (x)
- 3rd answer ⇒ y = 1 (x)
- 4th answer ⇒ x is seven units greater than y (√)
( x = 10 and y = 3 then x - y = 10 - 3 = 7)
- 5th answer ⇒ m∠ABC = 122° (√)
- 6th answer ⇒ PQ = 3 (x)
