4-6 Isosceles and Equilateral Triangles.pdf - Free Printable
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Step-by-step solution for: 4-6 Isosceles and Equilateral Triangles.pdf
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Step-by-step solution for: 4-6 Isosceles and Equilateral Triangles.pdf
This image is a geometry worksheet focused on "Isosceles and Equilateral Triangles." The task is to find the value of $x$ for each of the 10 problems presented.
To solve this, I will need to apply geometric principles related to these specific types of triangles:
- Isosceles Triangle: A triangle with at least two equal sides. The angles opposite the equal sides are also equal (Base Angles Theorem).
- Equilateral Triangle: A triangle with all three sides equal. All three interior angles are also equal, and each measures $60^\circ$.
- Triangle Sum Theorem: The sum of the interior angles of any triangle is always $180^\circ$.
I will now analyze each problem one by one, using these rules to find the value of $x$.
---
- This is an isosceles triangle with two sides marked as equal.
- The base angle is given as $68^\circ$. Since it's isosceles, the other base angle is also $68^\circ$.
- The vertex angle is labeled $x$.
- Using the Triangle Sum Theorem: $x + 68^\circ + 68^\circ = 180^\circ$
- $x + 136^\circ = 180^\circ$
- $x = 180^\circ - 136^\circ = 44^\circ$
Answer for 1: $x = 44^\circ$
---
- This is an isosceles triangle with two sides marked as equal.
- One of the base angles is given as $76^\circ$, so the other base angle is also $76^\circ$.
- The vertex angle is labeled $x$.
- $x + 76^\circ + 76^\circ = 180^\circ$
- $x + 152^\circ = 180^\circ$
- $x = 180^\circ - 152^\circ = 28^\circ$
Answer for 2: $x = 28^\circ$
---
- This is an equilateral triangle (all three sides are marked as equal).
- In an equilateral triangle, all angles are $60^\circ$.
- The angle labeled $x$ is one of the interior angles.
- Therefore, $x = 60^\circ$.
Answer for 3: $x = 60^\circ$
---
- This is an isosceles triangle with two sides marked as equal.
- The base angle is given as $30^\circ$, so the other base angle is also $30^\circ$.
- The vertex angle is labeled $x$.
- $x + 30^\circ + 30^\circ = 180^\circ$
- $x + 60^\circ = 180^\circ$
- $x = 180^\circ - 60^\circ = 120^\circ$
Answer for 4: $x = 120^\circ$
---
- This figure shows two intersecting lines forming two triangles.
- The lower triangle has a base angle of $78^\circ$ and is marked as isosceles (two sides are equal). So, the other base angle is also $78^\circ$.
- The vertex angle of this lower triangle can be found: $180^\circ - 78^\circ - 78^\circ = 24^\circ$.
- The vertex angle of the lower triangle and the angle labeled $x$ in the upper triangle are vertical angles. Vertical angles are equal.
- Therefore, $x = 24^\circ$.
Answer for 5: $x = 24^\circ$
---
- This is an isosceles triangle with a line segment from the vertex to the base, which appears to be the altitude and median (since it bisects the base and creates two right angles).
- The base is split into two segments, one labeled $9$ and the other labeled $x$.
- Because the triangle is isosceles and the line segment is drawn from the vertex to the midpoint of the base (indicated by the tick marks), the two segments must be equal.
- Therefore, $x = 9$.
Answer for 6: $x = 9$
---
- Similar to problem 6, this is an isosceles triangle with a line segment from the vertex to the base, creating two congruent right triangles.
- The base is split into two segments, one labeled $12$ and the other labeled $x$.
- The tick marks on the two base segments indicate they are equal.
- Therefore, $x = 12$.
Answer for 7: $x = 12$
---
- This figure consists of two triangles sharing a common side.
- The left triangle is isosceles with two sides marked as equal. The base angle is given as $68^\circ$, so the other base angle is also $68^\circ$.
- The vertex angle of the left triangle is: $180^\circ - 68^\circ - 68^\circ = 44^\circ$.
- The vertex angle of the left triangle and the angle labeled $x$ in the right triangle are vertical angles.
- Therefore, $x = 44^\circ$.
Answer for 8: $x = 44^\circ$
---
- This is an isosceles triangle with two sides marked as equal.
- The vertex angle is given as $60^\circ$.
- Since the triangle is isosceles, the two base angles are equal. Let each base angle be $y$.
- $60^\circ + y + y = 180^\circ$
- $60^\circ + 2y = 180^\circ$
- $2y = 120^\circ$
- $y = 60^\circ$
- The angle labeled $x$ is one of the base angles, so $x = 60^\circ$.
- Note: A triangle with all angles equal to $60^\circ$ is equilateral, which is consistent with the side markings.
Answer for 9: $x = 60^\circ$
---
- This is a right triangle that is also isosceles (two legs are marked as equal).
- The right angle is indicated by the square symbol.
- The two acute angles are equal because the triangle is isosceles.
- The sum of the angles in a triangle is $180^\circ$, so the two acute angles must sum to $90^\circ$.
- Since they are equal, each acute angle is $45^\circ$.
- The angle labeled $x$ is one of the acute angles.
- Therefore, $x = 45^\circ$.
Answer for 10: $x = 45^\circ$
---
Here is a complete list of the solutions for each problem:
1. $x = 44^\circ$
2. $x = 28^\circ$
3. $x = 60^\circ$
4. $x = 120^\circ$
5. $x = 24^\circ$
6. $x = 9$
7. $x = 12$
8. $x = 44^\circ$
9. $x = 60^\circ$
10. $x = 45^\circ$
To solve this, I will need to apply geometric principles related to these specific types of triangles:
- Isosceles Triangle: A triangle with at least two equal sides. The angles opposite the equal sides are also equal (Base Angles Theorem).
- Equilateral Triangle: A triangle with all three sides equal. All three interior angles are also equal, and each measures $60^\circ$.
- Triangle Sum Theorem: The sum of the interior angles of any triangle is always $180^\circ$.
I will now analyze each problem one by one, using these rules to find the value of $x$.
---
Problem 1
- This is an isosceles triangle with two sides marked as equal.
- The base angle is given as $68^\circ$. Since it's isosceles, the other base angle is also $68^\circ$.
- The vertex angle is labeled $x$.
- Using the Triangle Sum Theorem: $x + 68^\circ + 68^\circ = 180^\circ$
- $x + 136^\circ = 180^\circ$
- $x = 180^\circ - 136^\circ = 44^\circ$
Answer for 1: $x = 44^\circ$
---
Problem 2
- This is an isosceles triangle with two sides marked as equal.
- One of the base angles is given as $76^\circ$, so the other base angle is also $76^\circ$.
- The vertex angle is labeled $x$.
- $x + 76^\circ + 76^\circ = 180^\circ$
- $x + 152^\circ = 180^\circ$
- $x = 180^\circ - 152^\circ = 28^\circ$
Answer for 2: $x = 28^\circ$
---
Problem 3
- This is an equilateral triangle (all three sides are marked as equal).
- In an equilateral triangle, all angles are $60^\circ$.
- The angle labeled $x$ is one of the interior angles.
- Therefore, $x = 60^\circ$.
Answer for 3: $x = 60^\circ$
---
Problem 4
- This is an isosceles triangle with two sides marked as equal.
- The base angle is given as $30^\circ$, so the other base angle is also $30^\circ$.
- The vertex angle is labeled $x$.
- $x + 30^\circ + 30^\circ = 180^\circ$
- $x + 60^\circ = 180^\circ$
- $x = 180^\circ - 60^\circ = 120^\circ$
Answer for 4: $x = 120^\circ$
---
Problem 5
- This figure shows two intersecting lines forming two triangles.
- The lower triangle has a base angle of $78^\circ$ and is marked as isosceles (two sides are equal). So, the other base angle is also $78^\circ$.
- The vertex angle of this lower triangle can be found: $180^\circ - 78^\circ - 78^\circ = 24^\circ$.
- The vertex angle of the lower triangle and the angle labeled $x$ in the upper triangle are vertical angles. Vertical angles are equal.
- Therefore, $x = 24^\circ$.
Answer for 5: $x = 24^\circ$
---
Problem 6
- This is an isosceles triangle with a line segment from the vertex to the base, which appears to be the altitude and median (since it bisects the base and creates two right angles).
- The base is split into two segments, one labeled $9$ and the other labeled $x$.
- Because the triangle is isosceles and the line segment is drawn from the vertex to the midpoint of the base (indicated by the tick marks), the two segments must be equal.
- Therefore, $x = 9$.
Answer for 6: $x = 9$
---
Problem 7
- Similar to problem 6, this is an isosceles triangle with a line segment from the vertex to the base, creating two congruent right triangles.
- The base is split into two segments, one labeled $12$ and the other labeled $x$.
- The tick marks on the two base segments indicate they are equal.
- Therefore, $x = 12$.
Answer for 7: $x = 12$
---
Problem 8
- This figure consists of two triangles sharing a common side.
- The left triangle is isosceles with two sides marked as equal. The base angle is given as $68^\circ$, so the other base angle is also $68^\circ$.
- The vertex angle of the left triangle is: $180^\circ - 68^\circ - 68^\circ = 44^\circ$.
- The vertex angle of the left triangle and the angle labeled $x$ in the right triangle are vertical angles.
- Therefore, $x = 44^\circ$.
Answer for 8: $x = 44^\circ$
---
Problem 9
- This is an isosceles triangle with two sides marked as equal.
- The vertex angle is given as $60^\circ$.
- Since the triangle is isosceles, the two base angles are equal. Let each base angle be $y$.
- $60^\circ + y + y = 180^\circ$
- $60^\circ + 2y = 180^\circ$
- $2y = 120^\circ$
- $y = 60^\circ$
- The angle labeled $x$ is one of the base angles, so $x = 60^\circ$.
- Note: A triangle with all angles equal to $60^\circ$ is equilateral, which is consistent with the side markings.
Answer for 9: $x = 60^\circ$
---
Problem 10
- This is a right triangle that is also isosceles (two legs are marked as equal).
- The right angle is indicated by the square symbol.
- The two acute angles are equal because the triangle is isosceles.
- The sum of the angles in a triangle is $180^\circ$, so the two acute angles must sum to $90^\circ$.
- Since they are equal, each acute angle is $45^\circ$.
- The angle labeled $x$ is one of the acute angles.
- Therefore, $x = 45^\circ$.
Answer for 10: $x = 45^\circ$
---
Final Answers
Here is a complete list of the solutions for each problem:
1. $x = 44^\circ$
2. $x = 28^\circ$
3. $x = 60^\circ$
4. $x = 120^\circ$
5. $x = 24^\circ$
6. $x = 9$
7. $x = 12$
8. $x = 44^\circ$
9. $x = 60^\circ$
10. $x = 45^\circ$
Parent Tip: Review the logic above to help your child master the concept of isosceles and equilateral triangles worksheet.
