Solve for X, Y, and Z: Triangle Geometry with Given Angles 25°, 72°, and 105°

Triangle Angles with Exterior Angle Properties

Calculate the values of x, y, and z.

z25yx72105

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find angles Y,X,Z
00:05 Sum of angles in a triangle equals 180
00:09 Group terms and isolate Z
00:29 This is the value of angle Z
00:35 Supplementary angle to straight angle (180)
00:45 Isolate angle Y
00:53 This is the value of angle Y
01:01 Sum of angles in a triangle equals 180
01:08 Group terms and isolate X
01:25 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

Calculate the values of x, y, and z.

z25yx72105

2

Step-by-step solution

Angle Y complements 180 and we can calculate it since we know the adjacent angle.

Let's calculate it as follows:

180105=75 180-105=75

Now that we found angle Y, we can calculate angle X since we have the other two angles in the same triangle: 72 and 75.

1807572=33 180-75-72=33

We can calculate angle Z since we have two angles in the triangle: 25 and 105

The sum of angles in a triangle is 180, so we'll calculate Z as follows:

18010525=50 180-105-25=50

3

Final Answer

x = 33, y = 75, z = 50

Key Points to Remember

Essential concepts to master this topic
  • Triangle Sum: All three interior angles must add to 180°
  • Linear Pairs: Adjacent angles on a line sum to 180°: y = 180° - 105° = 75°
  • Check: Verify each triangle's angles sum to exactly 180° ✓

Common Mistakes

Avoid these frequent errors
  • Using exterior angles in triangle sum calculations
    Don't add exterior angle 105° with other triangle angles = wrong triangle sum! The 105° angle is outside the triangle, not inside. Always identify which angles are actually interior to each triangle before calculating.

Practice Quiz

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Indicates which angle is greater

FAQ

Everything you need to know about this question

How do I know which angles are interior vs exterior?

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Interior angles are inside the triangle shape. Exterior angles are formed when you extend a side beyond the triangle. In this problem, 105° is exterior because it's outside the triangle boundary.

Why does y equal 75° instead of 105°?

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Angles y and 105° form a linear pair on a straight line. Since they must add to 180°, we get: y=180°105°=75° y = 180° - 105° = 75°

Can I solve these angles in any order?

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Not always! Start with angles that have direct relationships first. Here, find y using the linear pair, then use y to find x, and finally find z.

What if my triangle angles don't add to 180°?

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Double-check your work! You may have confused interior and exterior angles or made a calculation error. Each triangle's interior angles must always sum to exactly 180°.

How do I identify all the triangles in this figure?

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  • Left triangle: angles x, 72°, and 75°
  • Right triangle: angles 25°, 105°, and z
  • The line through the middle creates these two separate triangles

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