In the diagram shown, PQRS is a rhombus and PQT is an isosceles triangle in which PT = QT.Angle PSR 110°. what is the size of angle SQT?

Let's analyze the problem step by step.

### Step 1: Understand the Given Information
- **PQRS** is a rhombus, which means all sides are equal, and opposite angles are equal.
- **PQT** is an isosceles triangle with \( PT = QT \).
- The measure of angle \( \angle PSR \) is given as \( 110^\circ \).

### Step 2: Find the Other Angles of the Rhombus
In a rhombus, opposite angles are equal, and adjacent angles are supplementary (they add up to \( 180^\circ \)). Given that \( \angle PSR = 110^\circ \), the angle \( \angle PQR \) opposite to \( \angle PSR \) is also \( 110^\circ \).

Since adjacent angles in a rhombus are supplementary:
\[
\angle PQS = 180^\circ - 110^\circ = 70^\circ
\]
Therefore, both \( \angle PQS \) and \( \angle QRP \) are \( 70^\circ \).

### Step 3: Analyze Triangle PQT
Given that \( PT = QT \) in the isosceles triangle \( PQT \), the base angles \( \angle PQT \) and \( \angle QPT \) are equal.

Let's denote the base angles as \( x \). Since \( \angle PQS = 70^\circ \), \( \angle PQT \) is the exterior angle for triangle \( PQT \) at vertex \( Q \). Therefore, the exterior angle theorem gives:
\[
\angle PQT = \angle P + \angle Q = x + x = 2x
\]
However, \( \angle PQS = 70^\circ \) is not the exterior angle directly affecting \( \angle SQT \), since \( SQT \) doesn't include \( \angle PQT \).

### Step 4: Identify \( \angle SQT \)
The angle \( \angle SQT \) is external to triangle \( PQT \). Knowing that the entire \( \angle PSR \) covers \( \angle PQS + \angle SQT \), we conclude:
\[
\angle PSR = \angle PQS + \angle SQT = 110^\circ
\]
Since \( \angle PQS \) is \( 70^\circ \):
\[
\angle SQT = \angle PSR - \angle PQS = 110^\circ - 70^\circ = 40^\circ
\]

### Conclusion
The size of angle \( \angle SQT \) is \( \boxed{40^\circ} \).

Would you like further details or have any questions?

Here are some related questions:
1. What is the sum of the angles in any quadrilateral?
2. How do you determine the area of a rhombus given its side lengths?
3. How does the interior angle of an equilateral triangle compare with that of a rhombus?
4. How would this problem change if the triangle \( PQT \) were equilateral?
5. What is the significance of isosceles triangles in different geometric proofs?

**Tip:** When dealing with rhombuses, remember that adjacent angles add up to \( 180^\circ \), which can help solve various angle-related problems.

Explore how to calculate angle SQT in a geometric problem involving a rhombus and an isosceles triangle. Understand the properties of rhombuses and isosceles triangles, including the use of the exterior angle theorem. Suitable for high school students studying geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation