Question

1. given δxps ≅ δdnf, find the values of x and y.
2. given δmtw ≅ δbgk, find the values of x and y.
3. given δprs ≅ δcfh, find the values of x, y, and z.

Explanation:

Step1: Use the property of congruent triangles (corresponding sides are equal) for problem 7

Since \(\triangle XPS\cong\triangle DNF\), then \(XS = ND\) and \(XP=NF\).
For \(XS = ND\):
\(17x + 3=57\)
\(17x=57 - 3\)
\(17x=54\)
\(x = 3\)
For \(XP = NF\):
\(4y-3 = 51\)
\(4y=51 + 3\)
\(4y=54\)
\(y = 13.5\)

Step2: Use the property of congruent triangles (corresponding angles are equal) for problem 8

Since \(\triangle MTW\cong\triangle BGK\), then \(\angle T=\angle G\) and \(\angle M=\angle B\)
For \(\angle T=\angle G\):
\(4x-3 = 45\)
\(4x=45 + 3\)
\(4x=48\)
\(x = 12\)
For \(\angle M=\angle B\):
In \(\triangle MTW\), \(\angle M=180^{\circ}-\angle T-\angle W=180^{\circ}-45^{\circ}-41^{\circ}=94^{\circ}\)
So \(11y + 6=94\)
\(11y=94 - 6\)
\(11y=88\)
\(y = 8\)

Step3: Use the property of congruent triangles (corresponding sides and angles are equal) for problem 9

Since \(\triangle PRS\cong\triangle CFH\), then \(PS = CH\), \(\angle R=\angle F\) (but \(\angle F = 90^{\circ}\)), \(\angle P=\angle C\) (but \(\angle C\) is not given in a form to use here), \(\angle S=\angle H\)
For \(PS = CH\):
\(2x-7=39\)
\(2x=39 + 7\)
\(2x=46\)
\(x = 23\)
For \(\angle R=\angle F\):
\(13y-1 = 90\)
\(13y=90 + 1\)
\(13y=91\)
\(y = 7\)
For \(\angle S=\angle H\):
In \(\triangle PRS\), \(\angle S=180^{\circ}-\angle P-\angle R=180^{\circ}-28^{\circ}-90^{\circ}=62^{\circ}\)
So \(6z-4 = 62\)
\(6z=62 + 4\)
\(6z=66\)
\(z = 11\)

Answer:

• Problem 7: \(x = 3\), \(y = 13.5\)
• Problem 8: \(x = 12\), \(y = 8\)
• Problem 9: \(x = 23\), \(y = 7\), \(z = 11\)