Find the base angles of an isosceles triangle if the vertex angle is a. 48° b. 70 ° 41' C. 102° 37' d. 30° 42' e. 56°38'46" f. 100° 24.8'

To find the base angles of an isosceles triangle when the vertex angle is given, we use the following steps:

1. **Recall the sum of the angles in any triangle** is always \( 180^\circ \).
2. **Subtract the vertex angle** from \( 180^\circ \) to find the sum of the two base angles.
3. **Divide the result by 2** to find each base angle, since they are equal in an isosceles triangle.

Let's calculate the base angles for each given vertex angle:

### a. Vertex angle \( 48^\circ \)

\[
\text{Sum of base angles} = 180^\circ - 48^\circ = 132^\circ
\]
\[
\text{Each base angle} = \frac{132^\circ}{2} = 66^\circ
\]

### b. Vertex angle \( 70^\circ 41' \)

First, subtract:
\[
180^\circ - 70^\circ 41' = 109^\circ 19'
\]
\[
\text{Each base angle} = \frac{109^\circ 19'}{2} = 54^\circ 39.5'
\]

### c. Vertex angle \( 102^\circ 37' \)

First, subtract:
\[
180^\circ - 102^\circ 37' = 77^\circ 23'
\]
\[
\text{Each base angle} = \frac{77^\circ 23'}{2} = 38^\circ 41.5'
\]

### d. Vertex angle \( 30^\circ 42' \)

First, subtract:
\[
180^\circ - 30^\circ 42' = 149^\circ 18'
\]
\[
\text{Each base angle} = \frac{149^\circ 18'}{2} = 74^\circ 39'
\]

### e. Vertex angle \( 56^\circ 38' 46'' \)

First, subtract:
\[
180^\circ - 56^\circ 38' 46'' = 123^\circ 21' 14''
\]
\[
\text{Each base angle} = \frac{123^\circ 21' 14''}{2} = 61^\circ 40' 37''
\]

### f. Vertex angle \( 100^\circ 24.8' \)

First, subtract:
\[
180^\circ - 100^\circ 24.8' = 79^\circ 35.2'
\]
\[
\text{Each base angle} = \frac{79^\circ 35.2'}{2} = 39^\circ 47.6'
\]

### Summary of the base angles:
1. **48°**: Each base angle = 66°
2. **70° 41'**: Each base angle = 54° 39.5'
3. **102° 37'**: Each base angle = 38° 41.5'
4. **30° 42'**: Each base angle = 74° 39'
5. **56° 38' 46"**: Each base angle = 61° 40' 37"
6. **100° 24.8'**: Each base angle = 39° 47.6'

Would you like any further details or have any questions?

Here are some related questions:
1. How do you convert degrees to radians for these angles?
2. How would you find the side lengths given one side in an isosceles triangle?
3. Can you solve for the altitude of the isosceles triangle in each case?
4. How does changing the vertex angle affect the base angles?
5. What are the properties of an equilateral triangle, a special case of an isosceles triangle?

**Tip:** Always ensure angle units are consistent when calculating (degrees, minutes, and seconds).

Learn how to calculate the base angles of an isosceles triangle when the vertex angle is given. Includes step-by-step solutions for vertex angles of 48°, 70° 41', 102° 37', 30° 42', 56° 38' 46", and 100° 24.8'. Explore the properties of triangles and theorems related to angles.

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