https://file.aiquickdraw.com/mathbot/file/17267403584755annku6h.jpg

To find the size of angle \( r \), we can use the properties of angles in triangles and around straight lines.

### Step-by-step solution:

1. **Angles on a straight line**:
   The angle labeled \( 112^\circ \) and the adjacent angle (the exterior angle at the triangle's vertex) are supplementary, meaning their sum is \( 180^\circ \) because they form a straight line. Thus, the exterior angle is:
   \[
   180^\circ - 112^\circ = 68^\circ
   \]

2. **Sum of angles in a triangle**:
   The interior angles of a triangle must sum to \( 180^\circ \). The three angles in the triangle are:
   - \( 46^\circ \) (given),
   - \( r \) (unknown), and
   - the exterior angle we just found, which is \( 68^\circ \).

   Using the sum of interior angles:
   \[
   r + 46^\circ + 68^\circ = 180^\circ
   \]

3. **Solve for \( r \)**:
   Simplifying the equation:
   \[
   r + 114^\circ = 180^\circ
   \]
   \[
   r = 180^\circ - 114^\circ = 66^\circ
   \]

Thus, the size of angle \( r \) is \( \boxed{66^\circ} \).

---

Would you like more details or have any questions? Here are five related questions for further understanding:

1. How would the solution change if the exterior angle was different?
2. Can you explain why the sum of angles in a triangle is always \( 180^\circ \)?
3. How would you find the exterior angle of a polygon with more than three sides?
4. What are the properties of supplementary angles on a straight line?
5. How can this method be applied to non-right triangles?

**Tip:** Always remember the sum of the angles in any triangle is \( 180^\circ \), which is a key property used to solve for unknown angles.

Find the size of angle r in the triangle, given that one angle is 46° and an exterior angle is 112°.

Learn how to find the unknown angle r in a triangle using the properties of supplementary angles and the sum of angles in a triangle. This geometry problem involves given angles of 46° and 112°, and is suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation