We are tasked with finding the degree measures of each angle in the isosceles triangle $QRS$. Here's the step-by-step breakdown:

Step 1: Identify given information

• Triangle $QRS$ is isosceles with base $QR$, so $\angle Q$ and $\angle R$ are the base angles and are equal: $m\angle Q = m\angle R.$
• We are given: $m\angle Q = (2x + 34)^\circ \quad \text{and} \quad m\angle R = (5x + 10)^\circ.$

Step 2: Use equality of base angles

Since $\angle Q = \angle R$, set the two expressions for $m\angle Q$ and $m\angle R$ equal: $2x + 34 = 5x + 10.$

Step 3: Solve for $x$

1. Subtract $2x$ from both sides: $34 = 3x + 10.$
2. Subtract $10$ from both sides: $24 = 3x.$
3. Divide by $3$: $x = 8.$

Step 4: Find $m\angle Q$ and $m\angle R$

Substitute $x = 8$ into the expressions for $m\angle Q$ and $m\angle R$:

1. For $m\angle Q = 2x + 34$: $m\angle Q = 2(8) + 34 = 16 + 34 = 50^\circ.$

2. For $m\angle R = 5x + 10$: $m\angle R = 5(8) + 10 = 40 + 10 = 50^\circ.$

Thus: $m\angle Q = m\angle R = 50^\circ.$

Step 5: Find $m\angle S$ (the vertex angle)

In a triangle, the sum of the interior angles is $180^\circ$. Therefore: $m\angle Q + m\angle R + m\angle S = 180^\circ.$ Substitute $m\angle Q = 50^\circ$ and $m\angle R = 50^\circ$: $50 + 50 + m\angle S = 180.$ Simplify: $m\angle S = 180 - 100 = 80^\circ.$

Final Answer:

The measures of the angles in $\triangle QRS$ are:

• $m\angle Q = 50^\circ$,
• $m\angle R = 50^\circ$,
• $m\angle S = 80^\circ$.

Summary Table:

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Related Questions:

1. What are the properties of isosceles triangles?
2. How do you solve systems of equations in geometry problems?
3. What is the sum of interior angles of any polygon?
4. Can you prove why the base angles of an isosceles triangle are always equal?
5. How do you find the missing angles in a triangle given two angles?

Tip:

In an isosceles triangle, always check for equality of base angles to simplify your calculations.