# Triangle and Rectangle

Pro Problems > Math > Geometry > Triangles## 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.

## Solution

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### 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

### 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?

### 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?

### 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 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?

### Three Two-Digit Angle Measures

The measures of the angles in a triangle are all two digit numbers, and two of them end in 3. Exactly two of them contain the digit 4. None of the angle measures are multiples of three. What are the measures of the angles?

### Triangle in a Square

Triangle ABC is inside square ABDE, and C lies along side DE. If the area of the triangle is 24 square units, what is the length of AB?

### 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.

### Triangle in a Square

If the perimeter of the triangle (in linear units) is equal to the area of the triangle (in square units), what is the length of the side of the square?

### Three Triangles

The angles in a triangle have measures x, x + 10, and y.

The angles in a second triangle have measures x + y, 40, and 50.

The angles in a third triangle have measures x – y, 40, and z.

What is the value of z?