Question

(10 pts) use the figure below to answer the following questions

a. list the two sides and the included angle of each triangle that are congruent to each other so that you can use sas theorem.
b. write a congruence statement for the two triangles.
c. solve for \\(x\\) and find \\(m\angle cbd\\).

Explanation:

Identify congruent parts for SAS

We analyze the given figure to find congruent parts in \(\triangle ACD\) and \(\triangle BCD\).

• Side \(AC = BC = 14\).
• Angle \(\angle ACD = \angle BCD = 45^\circ\).
• Side \(CD\) is shared by both triangles (Reflexive Property), so \(CD = CD\).

These parts form the Side-Angle-Side (SAS) configuration.

Write the congruence statement

Using the congruent parts identified:

• Vertex \(A\) corresponds to vertex \(B\).
• Vertex \(C\) corresponds to vertex \(C\).
• Vertex \(D\) corresponds to vertex \(D\).

Thus, the congruence statement is \(\triangle ACD \cong \triangle BCD\).

Set up equation for x

Since \(\triangle ACD \cong \triangle BCD\), their corresponding angles are equal by CPCTC.
Therefore, \(\angle CAD = \angle CBD\).
From the figure:

• \(m\angle CAD = 15x\)
• \(m\angle CBD = 5x^2\)

We set up the equation:

$$5x^2 = 15x$$

Solve for x

We solve the quadratic equation:

$$5x^2 - 15x = 0$$

$$5x(x - 3) = 0$$

This gives two possible solutions:

• \(x = 0\) (not geometrically possible as angle measures would be \(0^\circ\))
• \(x = 3\)

Thus, \(x = 3\).

Find the measure of angle CBD

We substitute \(x = 3\) back into the expression for \(m\angle CBD\):

$$m\angle CBD = 5(3)^2 = 5(9) = 45^\circ$$

Answer:

Question A

The congruent parts for the SAS theorem are:

• Sides: \(AC \cong BC\) and \(CD \cong CD\)
• Included Angles: \(\angle ACD \cong \angle BCD\)

Question B

The congruence statement is:

$$\triangle ACD \cong \triangle BCD$$

Question C

The value of \(x\) and the angle measure are:

• \(x = 3\)
• \(m\angle CBD = 45^\circ\)