We are given a diagram involving angles formed by a straight line and a ray, with angle measures expressed in terms of $ x $. Let's analyze the image and solve for $ x $.
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Step 1: Understand the Diagram
We have:
- A straight line $ XZ $ with point $ Y $ on it.
- A ray $ YO $ extending upward from point $ Y $, forming two angles at point $ Y $:
- One angle is labeled $ (3x + 5)^\circ $
- The other is labeled $ (2x - 5)^\circ $
These two angles are adjacent and form a
linear pair because they lie on a straight line ($ XZ $) and share a common vertex and side. Therefore, their sum must be $ 180^\circ $.
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Step 2: Set Up the Equation
Since the angles form a linear pair:
$$
(3x + 5) + (2x - 5) = 180
$$
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Step 3: Simplify and Solve
$$
3x + 5 + 2x - 5 = 180
$$
$$
5x = 180
$$
$$
x = \frac{180}{5} = 36
$$
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Step 4: Find the Angle Measures
Now substitute $ x = 36 $ into each expression:
- First angle: $ 3x + 5 = 3(36) + 5 = 108 + 5 = 113^\circ $
- Second angle: $ 2x - 5 = 2(36) - 5 = 72 - 5 = 67^\circ $
Check: $ 113^\circ + 67^\circ = 180^\circ $
✔
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✔ Final Answer:
$$
\boxed{x = 36}
$$
The two angles are $ 113^\circ $ and $ 67^\circ $, which add up to $ 180^\circ $, confirming the solution is correct.
Parent Tip: Review the logic above to help your child master the concept of angle addition postulate worksheet.
