Question

find the value of each variable if ( \triangle lmncong\triangle pqr ).

1. ( mangle l = 35^{circ};mangle p = 5x^{circ} )
2. ( mangle m = 2a^{circ};mangle q=(3a - 20)^{circ} )
3. ( mn=x + 45;qr = 4x-30 )
4. ( ln=r - 2;pr = 2r-14 )

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle LMN\cong\triangle PQR\), corresponding angles and sides are equal. For problem 27, \(\angle L\) and \(\angle P\) are corresponding angles. So \(m\angle L = m\angle P\).
\(35 = 5x\)

Step2: Solve for \(x\)

Divide both sides of the equation \(35 = 5x\) by \(5\).
\(x=\frac{35}{5}=7\)

For problem 28, \(\angle M\) and \(\angle Q\) are corresponding angles. So \(m\angle M = m\angle Q\).
\(2a=3a - 20\)

Step3: Solve for \(a\)

Subtract \(2a\) from both sides: \(0=a - 20\). Then add \(20\) to both sides: \(a = 20\)

For problem 29, \(MN\) and \(QR\) are corresponding sides. So \(MN=QR\).
\(x + 45=4x-30\)

Step4: Solve for \(x\)

Subtract \(x\) from both sides: \(45 = 3x-30\). Add \(30\) to both sides: \(75=3x\). Divide by \(3\): \(x = 25\)

For problem 30, \(LN\) and \(PR\) are corresponding sides. So \(LN=PR\).
\(r-2=2r - 14\)

Step5: Solve for \(r\)

Subtract \(r\) from both sides: \(- 2=r - 14\). Add \(14\) to both sides: \(r=12\)

Answer:

1. \(x = 7\)
2. \(a = 20\)
3. \(x = 25\)
4. \(r = 12\)