Question

1. 14x - 5

6.
8.
9.
x =__ (14x - 5) =x = (15x) =__
x = ∠(6x - 12) =__y = ∠(3y) =__

Explanation:

5.

Step1: Set up the equation for the isosceles triangle

In an isosceles triangle, the two equal - side expressions are equal. So, \(14x−5 = 4x + 25\).

Step2: Solve for \(x\)

Subtract \(4x\) from both sides: \(14x−4x−5=4x−4x + 25\), which gives \(10x−5 = 25\).
Add \(5\) to both sides: \(10x−5 + 5=25 + 5\), so \(10x=30\).
Divide both sides by \(10\): \(x=\frac{30}{10}=3\).

Step3: Find the value of \(14x−5\)

Substitute \(x = 3\) into \(14x−5\): \(14\times3−5=42−5 = 37\).

6.

Step1: Set up the equation for the congruent triangles (they are right - isosceles and congruent, so their corresponding sides are equal)

\(15x=3x + 46+4x\).

Step2: Simplify the right - hand side

\(15x=7x + 46\).

Step3: Solve for \(x\)

Subtract \(7x\) from both sides: \(15x−7x=7x−7x + 46\), so \(8x=46\).
Divide both sides by \(8\): \(x=\frac{46}{8}=\frac{23}{4}=5.75\).

Step4: Find the value of \(15x\)

Substitute \(x = 5.75\) into \(15x\): \(15\times5.75 = 86.25\).

8.

Step1: Find the measure of \(\angle E\) in \(\triangle DEF\)

Using the triangle - angle sum theorem (\(\angle D+\angle E+\angle F = 180^{\circ}\)), \(\angle E=180-(36 + 67)=77^{\circ}\).

Step2: Set up the equation for the congruent angles (assuming \(\triangle DEF\cong\triangle LJK\))

\(6x−12 = 36\).

Step3: Solve for \(x\)

Add \(12\) to both sides: \(6x−12 + 12=36+12\), so \(6x=48\).
Divide both sides by \(6\): \(x = 8\).

Step4: Find the value of \(6x−12\)

Substitute \(x = 8\) into \(6x−12\): \(6\times8−12=48−12 = 36\).

9.

Step1: Use the property of right - triangles (the two non - right angles in a right - triangle are complementary)

\(3y+21 = 90\).

Step2: Solve for \(y\)

Subtract \(21\) from both sides: \(3y+21−21=90−21\), so \(3y=69\).
Divide both sides by \(3\): \(y = 23\).

Step3: Find the value of \(3y\)

Substitute \(y = 23\) into \(3y\): \(3\times23=69^{\circ}\).

Answer:

1. \(x = 3\), \((14x−5)=37\)
2. \(x = 5.75\), \((15x)=86.25\)
3. \(x = 8\), \((6x−12)=36^{\circ}\)
4. \(y = 23\), \(\angle(3y)=69^{\circ}\)