Question

10). given ( mangle yxz=(4x + 12)^{circ} ), and ( mangle xzy=(5x - 3)^{circ} ). use the triangle below to answer the following. show your work for credit.
11). solve for x:
12). list the (5) triangle congruence (abbreviations).
13). what does cpctc mean? write out what the acronym means - spelling counts!
14). in the figure below, ( w ) is the midpoint of ( overline{tv} ) and ( overline{uw}perpoverline{tv} ). mark the triangle with congruence marks and state which congruence theorem proves that ( \triangle uvwcong\triangle utw )?
15). write a congruence statement and that justifies the triangles congruent by sas.

Explanation:

Step1: Identify the triangle type

The triangle is isosceles (\(\overline{XY}\cong\overline{XZ}\)), so \(m\angle Y = m\angle Z\).

Step2: Use the triangle - angle sum theorem

\(m\angle Y+m\angle Z + m\angle X=180^{\circ}\). Since \(m\angle Y = m\angle Z\), we have \((4x + 12)+(5x-3)+(5x - 3)=180\).

Step3: Simplify the equation

Combine like terms: \(4x+12 + 5x-3+5x - 3=180\), \(14x+6 = 180\).

Step4: Solve for \(x\)

Subtract 6 from both sides: \(14x=180 - 6=174\), then \(x=\frac{174}{14}=12\).

Step5: Find \(m\angle X\)

\(m\angle X=(4x + 12)^{\circ}\), substitute \(x = 12\): \(m\angle X=4\times12+12=60^{\circ}\).

Step6: Find \(m\angle Y\) and \(m\angle Z\)

\(m\angle Y=m\angle Z=(5x - 3)^{\circ}\), substitute \(x = 12\): \(m\angle Y=m\angle Z=5\times12-3 = 57^{\circ}\).

Answer:

\(x = 12\)
\(m\angle X=60^{\circ}\)
\(m\angle Y = 57^{\circ}\)
\(m\angle Z=57^{\circ}\)