# Arithmetic Angles

Pro Problems > Math > Geometry > Triangles## Arithmetic Angles

The angles of a triangle are in arithmetic progression, and the difference between the largest and smallest angle is 42 degrees. What are the three angles?

## Solution

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### Angle Measure

Line segment AB is intersected by segment DC, with D on AB, between A and B.

In triangle ADC, m∠ADC = 30º + x and m∠DAC = 50º

In triangle ABC, m∠ACB = 115º + x

In triangle CDB, m∠CDB = 140º + x

Find m∠CBD.

### Angle Bisector

In the diagram shown,

m ∠DCF = 80

m ∠ACF = 30

m ∠FBD = 50.

If segment BF bisects ∠ABD, find m ∠BAC.

### Angles in a Triangle

In a triangle, the measure of the largest angle is 12 less than the sum of the measures of the other two angles. The largest angle is also 52 more than the twice the middle angle decreased by three times the smallest angle. What is the largest angle measure?

### Triangle and Rectangle

In the diagram displayed, the triangle splits the rectangle into three similar triangles. Find the value of AD^{2} divided by the product of DE and EC.

### Quadratic Triangle

The angles in a triangle have measures x^{2} - 5, 2x + 18, and x + 37. What is the measure of the largest angle in the triangle?

### Angle Riddle

The angles of a triangle meet the following criteria:

- All the angle measures are prime numbers.
- All the angle measures are distinct.
- Exactly two of my angle measures are palindromic.
- The difference between my two largest angle measures is a two digit number x.
- One of the digits of x is twice the other digit.

### Triangle Variables

An isosceles triangle is a triangle in which two of the angles have the same measure. In an isosceles triangle, the angle measures are 30 + x, 62 - x, and 10 + 2x + y. What are the possible numeric values for the measures of the angles?

### Two Triangles with Angle Measures

In a triangle, the angles measures are: 3x + 44, x, and 2x - 20. In a second triangle, two of the angle measures are x + 20 and 2x - 5. What is the measure of the third angle?

### Split Triangle

Segment AD is drawn in triangle ABC, with D on side BC. Given the following information, determine the measure of angle ADB.

measure of angle BAD = x

measure of angle CAD = 2x

measure of angle ADC = 2y

measure of angle ABD = y

measure of angle ACD = x + y

### Forty-five and Thirty

Two of the angles of a triangle are 45 degrees and 30 degrees. The altitude from the third angle has length 15 units. What is the triangle's area?