Answer:
x = 7
y = 37
Explanation:
Congruent: If two figures are congruent, they have the same shape and the same size.
Therefore, if the given triangles are congruent, the measures of their angles and sides will be the same (equal).
To find the value of x that will result in ΔLMN≅ΔPQR, equate the measure of angle P and angle L, then solve for x:
⇒ m∠P = m∠L
⇒ (7x - 4)° = (4x + 17)°
⇒ 7x - 4 = 4x + 17
⇒ 7x - 4 + 4 = 4x + 17 + 4
⇒ 7x = 4x + 21
⇒ 7x - 4x = 4x + 21 - 4x
⇒ 3x = 21
⇒ 3x ÷ 3 = 21 ÷ 3
⇒ x = 7
Substitute the found value of x into the expression for angle L to find m∠L:
⇒ m∠L = (4(7) + 17)°
⇒ m∠L = (28 + 17)°
⇒ m∠L = 45°
Interior angles of a triangle sum to 180°.
Therefore, to find the value of y, sum the angles of ΔLMN and equate to 180° to find the measure of angle N:
⇒ m∠L + m∠M + m∠N = 180°
⇒ 45° + 84° + (2x + y)° = 180°
⇒ 129° + (2x + y)° = 180°
⇒ 129° + (2x + y)° - 129° = 180° - 129°
⇒ (2x + y)° = 51°
⇒ 2x + y = 51
Substitute the found value of x and solve for y:
⇒ 2(7) + y = 51
⇒ 14 + y = 51
⇒ 14 + y - 14 = 51 - 14
⇒ y = 37
Conclusion
The value of x that will result in congruent triangles is
.
The value of y that will result in congruent triangle is
.