Question

Use the given information and the diagrams shown to determine the value of x.
Given: ∆LBR ≅ ∆MDS.
Angle L=45 Angle R =2x-1 & Angle S =115

Answer 1

To find the value of \( x \), we can utilize the information given about the congruent triangles \( \Delta LBR \) and \( \Delta MDS \). Since these triangles are congruent, their corresponding angles are equal.

From the information provided, we can summarize the angles in both triangles:

- In \( \Delta LBR \):
- \( \angle L = 45^\circ \)
- \( \angle R = 2x - 1^\circ \)

- In \( \Delta MDS \):
- \( \angle S = 115^\circ \)

Because the triangles are congruent, we can set up an equation that relates the angles. Since the angles in a triangle sum up to \( 180^\circ \), we can find the missing angle in \( \Delta LBR \):

\[
\angle B = 180^\circ - \angle L - \angle R
\]

Calculating \( \angle B \):

\[
\angle B = 180^\circ - 45^\circ - (2x - 1) = 180^\circ - 45^\circ - 2x + 1 = 136^\circ - 2x
\]

For triangle \( \Delta MDS \), we also need to find the angle \( D \):

\[
\angle D = 180^\circ - \angle S - \angle M
\]

Since \( \angle S = 115^\circ \) and assuming that angle \( M \) corresponds to \( \angle B \) (since \( \Delta LBR \) is congruent to \( \Delta MDS \)), we can conclude:

Setting \( \angle B \) equal to \( \angle D \):

\[
136^\circ - 2x = 180^\circ - 115^\circ - \angle M
\]

To simplify further, if we assume angle \( M \) is equal to angle \( L (45^\circ)\):

Substituting gives:

\[
136 - 2x = 180 - 115 - 45
\]
\[
136 - 2x = 20
\]
Rearranging gives:

\[
116 = 2x
\]
Dividing both sides by 2:

\[
x = 58
\]

Thus, the value of \( x \) is \( \boxed{58} \).