To solve the problem, we need to determine the value of \( x \) in each diagram using geometric principles, particularly properties of triangles and parallel lines. Let's analyze each diagram step by step.
---
Diagram 1:
- The top angle is \( 50^\circ \).
- The triangle is isosceles (indicated by the two equal sides marked with a single tick).
- In an isosceles triangle, the base angles are equal.
- Let the base angles be \( x \).
Using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
x + x + 50^\circ = 180^\circ
\]
\[
2x + 50^\circ = 180^\circ
\]
\[
2x = 130^\circ
\]
\[
x = 65^\circ
\]
So, for Diagram 1:
\[
x = 65^\circ
\]
---
Diagram 2:
- The top angle is \( 50^\circ \).
- The triangle is isosceles (indicated by the two equal sides marked with a single tick).
- The line segment drawn inside the triangle is perpendicular to the base, splitting the triangle into two right triangles.
- In each right triangle, the angle opposite the hypotenuse is \( 90^\circ \), and one of the angles is \( 50^\circ / 2 = 25^\circ \) (since the perpendicular bisects the vertex angle).
The remaining angle \( x \) in each right triangle is:
\[
x = 90^\circ - 25^\circ = 65^\circ
\]
So, for Diagram 2:
\[
x = 65^\circ
\]
---
Diagram 3:
- The triangle has two angles marked as \( 50^\circ \).
- Since the triangle has two equal angles, it is isosceles.
- The third angle \( x \) can be found using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
x + 50^\circ + 50^\circ = 180^\circ
\]
\[
x + 100^\circ = 180^\circ
\]
\[
x = 80^\circ
\]
So, for Diagram 3:
\[
x = 80^\circ
\]
---
Diagram 4:
- The triangle has two angles marked as \( 50^\circ \).
- The third angle \( x \) can be found using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
x + 50^\circ + 50^\circ = 180^\circ
\]
\[
x + 100^\circ = 180^\circ
\]
\[
x = 80^\circ
\]
So, for Diagram 4:
\[
x = 80^\circ
\]
---
Diagram 5:
- The triangle has one angle marked as \( 50^\circ \).
- The other angle is given as \( 65^\circ \).
- The third angle \( x \) can be found using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
x + 50^\circ + 65^\circ = 180^\circ
\]
\[
x + 115^\circ = 180^\circ
\]
\[
x = 65^\circ
\]
So, for Diagram 5:
\[
x = 65^\circ
\]
---
Diagram 6:
- The triangle has one angle marked as \( 50^\circ \).
- The other angle is given as \( 65^\circ \).
- The third angle \( x \) can be found using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
x + 50^\circ + 65^\circ = 180^\circ
\]
\[
x + 115^\circ = 180^\circ
\]
\[
x = 65^\circ
\]
So, for Diagram 6:
\[
x = 65^\circ
\]
---
Final Answers:
\[
\boxed{65^\circ, 65^\circ, 80^\circ, 80^\circ, 65^\circ, 65^\circ}
\]
Parent Tip: Review the logic above to help your child master the concept of parallel lines and triangles worksheet.
