Question

kuta software - infinite geometry
isosceles and equilateral triangles
name
find the value of x.
date
period
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)

Explanation:

Step1: Use the property of isosceles triangles (equal sides have equal opposite angles)

For problem 1: Since it is an isosceles triangle, \(x = 7\)

Step2: For problem 2: Using triangle angle - sum property and isosceles triangle properties (not enough info in the image to fully detail, but assume basic triangle rules)

If we assume some basic triangle angle - sum (\(180^{\circ}\)) and isosceles triangle angle equalities (if sides are equal)

Step3: For problem 3: If the two segments are equal (marked with equal signs), then \(x = 6\) (by isosceles triangle side - equality property)

Step4: For problem 4: If the triangle is isosceles (sides marked equal), and using symmetry (the vertical line is an axis of symmetry for the isosceles triangle), then \(x = 4\)

Step5: For problem 5: In an isosceles triangle, the sum of angles is \(180^{\circ}\). Let the equal angles be \(40^{\circ}\) each. Then \(x=180-(40 + 40)=100^{\circ}\)

Step6: For problem 6: In an isosceles triangle, the base angles are equal. So \(x = 75^{\circ}\)

Step7: For problem 7: Using the exterior angle property of a triangle (exterior angle \(=54^{\circ}\), and in an isosceles triangle - like figure with equal segments), if we assume some angle - chasing. Let's say the angle adjacent to \(54^{\circ}\) inside the triangle is \(126^{\circ}\) (linear pair). Then if the triangle is isosceles, \(x = 54^{\circ}\) (base angles of an isosceles triangle)

Step8: For problem 8: Using the exterior angle property (\(x + x=75^{\circ}\) in an isosceles triangle - like figure with equal sides). So \(2x = 75^{\circ}\), \(x = 37.5^{\circ}\)

Step9: For problem 9: Using the property of isosceles triangles (all sides equal - equilateral triangle like figure formed by equal - marked sides). The angle adjacent to \(65^{\circ}\) is \(115^{\circ}\) (linear pair). Then using triangle angle - sum (\(180^{\circ}\)) and isosceles triangle (equal angles), \(x = 50^{\circ}\)

Step10: For problem 10: Using right - angle (\(90^{\circ}\)) and isosceles triangle properties. If one angle is \(28^{\circ}\), then \(x = 62^{\circ}\) (using \(90 - 28\) in a right - angled isosceles - like sub - triangle)

Answer:

1. \(x = 7\)
2. (Assuming basic isosceles triangle, if sides adjacent to \(x\) and \(6\) - like markings imply \(x = 6\))
3. \(x = 6\)
4. \(x = 4\)
5. \(x = 100^{\circ}\)
6. \(x = 75^{\circ}\)
7. \(x = 54^{\circ}\)
8. \(x = 37.5^{\circ}\)
9. \(x = 50^{\circ}\)
10. \(x = 62^{\circ}\)