ΔLMN ≅ ΔPQR
LM = PQ, LN = PR, MN = QR
∠P = 55°, ∠Q = (y - 7)° find x +y


Given that \( \Delta LMN \cong \Delta PQR \), this means the triangles are congruent, and corresponding sides and angles are equal. 

You are given:
- \( \text{LM} = \text{PQ}, \text{LN} = \text{PR}, \text{MN} = \text{QR} \)
- \( \angle P = 55^\circ \)
- \( \angle Q = (y - 7)^\circ \)

Since \( \Delta LMN \cong \Delta PQR \), we also know the angles are congruent:
- \( \angle L \cong \angle P \)
- \( \angle M \cong \angle Q \)
- \( \angle N \cong \angle R \)

Thus, \( \angle L = 55^\circ \) and \( \angle M = (y - 7)^\circ \).

The sum of the angles in any triangle is 180°. For \( \Delta PQR \), this gives:
\[
\angle P + \angle Q + \angle R = 180^\circ
\]
Substituting the known values:
\[
55^\circ + (y - 7)^\circ + \angle R = 180^\circ
\]
Simplify:
\[
55^\circ + y - 7^\circ + \angle R = 180^\circ
\]
\[
48^\circ + y + \angle R = 180^\circ
\]
\[
y + \angle R = 132^\circ
\]
\[
\angle R = 132^\circ - y
\]

Since \( \angle N \cong \angle R \), and the sum of the angles in \( \Delta LMN \) is also 180°:
\[
55^\circ + (y - 7)^\circ + (132^\circ - y) = 180^\circ
\]
Simplifying:
\[
55^\circ + y - 7^\circ + 132^\circ - y = 180^\circ
\]
\[
180^\circ = 180^\circ
\]
This is a valid equation, meaning the relationship holds.

To find \( x + y \), we assume \( x \) is equal to the measure of a corresponding angle from either triangle. In this case, \( \angle N = \angle R = 132^\circ - y \). Therefore, \( x = 132^\circ - y \), and to find \( x + y \):
\[
x + y = (132^\circ - y) + y = 132^\circ
\]

Thus, \( x + y = 132^\circ \).

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. If \( \angle R = 60^\circ \), what is the value of \( y \)?
2. What is the sum of angles in any triangle?
3. How can you verify that two triangles are congruent?
4. Given two congruent triangles, how are their corresponding sides and angles related?
5. How do you calculate the third angle in a triangle if two angles are known?

**Tip:** When dealing with congruent triangles, remember that corresponding parts (sides and angles) are always equal.

ΔLMN ≅ ΔPQR
LM = PQ, LN = PR, MN = QR
∠P = 55°, ∠Q = (y - 7)° find x + y

Learn how to solve a problem involving congruent triangles ΔLMN and ΔPQR. This question requires understanding congruent triangles, the triangle angle sum theorem, and basic algebra. The solution explores how to find x + y using given angles and side lengths. Suitable for Grades 9-10.

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