Math / Geometry

Question9. ${ }^{7}$ (a) If the two angles of a triangle are $50^{\circ}$ and $75^{\circ}$ , find the third ang. b) If $2 x^{\circ}, 4 x^{\circ}$ and $6 x^{\circ}$ are the three angles of a triangle, find the valuer is find the angles. (c) If the three angles of a triangle are $x^{\circ},(x+20)^{\circ}$ and $(x-50)^{\circ}$ , find x . Also, find the angles. (d) If the two opposite interior angles of a triangle are $30^{\circ}$ and $55^{\circ}$ exterior angle of a triangle.
10. त(3) In a right angled triangle, one acute angle is $40^{\circ}$ . Find the other atexes that triangle. (b) In an equilateral triangle, one angle is $60^{\circ}$ . Find the remaining anglo (c) In an isosceles triangle, one base angle is $45^{\circ}$ . Find the another bases
स'(d) If one angle of an isosceles triangle is equal to the sum of remaining angles, find the size of each angle.
11. (a) In a $\triangle A B C, \angle A=\angle B=65^{\circ}$ , find $\angle C$ .

Studdy Solution

STEP 1

1. We are dealing with various types of triangles.
2. The sum of angles in any triangle is $180^\circ$ .
3. For equilateral triangles, all angles are $60^\circ$ .
4. For isosceles triangles, two angles are equal.
5. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

STEP 2

1. Solve problem 9(a) by finding the third angle of a triangle.
2. Solve problem 9(b) by finding the value of $x$ for given angles.
3. Solve problem 9(c) by finding $x$ and the angles of the triangle.
4. Solve problem 9(d) by finding the exterior angle.
5. Solve problem 10(a) by finding the other acute angle in a right triangle.
6. Solve problem 10(b) by confirming the angles of an equilateral triangle.
7. Solve problem 10(c) by finding the other base angle in an isosceles triangle.
8. Solve problem 10(d) by finding the angles of an isosceles triangle where one angle is the sum of the others.
9. Solve problem 11(a) by finding the third angle in an isosceles triangle.

STEP 3

For problem 9(a), we are given two angles of a triangle: $50^\circ$ and $75^\circ$ . To find the third angle, use the fact that the sum of angles in a triangle is $180^\circ$ .
$\text{Third angle} = 180^\circ - 50^\circ - 75^\circ$ $= 55^\circ$

STEP 4

For problem 9(b), the angles are $2x^\circ, 4x^\circ,$ and $6x^\circ$ . The sum of angles in a triangle is $180^\circ$ .
$2x + 4x + 6x = 180$ $12x = 180$ $x = 15$
The angles are $2x = 30^\circ, 4x = 60^\circ, 6x = 90^\circ$ .

STEP 5

For problem 9(c), the angles are $x^\circ, (x+20)^\circ, (x-50)^\circ$ . The sum of angles in a triangle is $180^\circ$ .
$x + (x+20) + (x-50) = 180$ $3x - 30 = 180$ $3x = 210$ $x = 70$
The angles are $70^\circ, 90^\circ, 20^\circ$ .

STEP 6

For problem 9(d), the two opposite interior angles are $30^\circ$ and $55^\circ$ . The exterior angle is the sum of these two angles.
$\text{Exterior angle} = 30^\circ + 55^\circ = 85^\circ$

STEP 7

For problem 10(a), in a right-angled triangle, one acute angle is $40^\circ$ . The other acute angle is:
$90^\circ - 40^\circ = 50^\circ$

STEP 8

For problem 10(b), in an equilateral triangle, all angles are $60^\circ$ . Therefore, the remaining angles are also $60^\circ$ .

STEP 9

For problem 10(c), in an isosceles triangle, one base angle is $45^\circ$ . The other base angle is also $45^\circ$ .

STEP 10

For problem 10(d), if one angle of an isosceles triangle is equal to the sum of the remaining angles, then:
Let the equal angles be $x$ and the third angle be $2x$ .
$x + x + 2x = 180$ $4x = 180$ $x = 45$
The angles are $45^\circ, 45^\circ, 90^\circ$ .

STEP 11

For problem 11(a), in $\triangle ABC$ , $\angle A = \angle B = 65^\circ$ . Find $\angle C$ :
$\angle C = 180^\circ - 65^\circ - 65^\circ$ $= 50^\circ$

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Studdy Solution

STEP 1

STEP 2

STEP 3

For problem 9(a), we are given two angles of a triangle: $50^\circ$ and $75^\circ$ . To find the third angle, use the fact that the sum of angles in a triangle is $180^\circ$ .
$\text{Third angle} = 180^\circ - 50^\circ - 75^\circ$ $= 55^\circ$

STEP 4

For problem 9(b), the angles are $2x^\circ, 4x^\circ,$ and $6x^\circ$ . The sum of angles in a triangle is $180^\circ$ .
$2x + 4x + 6x = 180$ $12x = 180$ $x = 15$
The angles are $2x = 30^\circ, 4x = 60^\circ, 6x = 90^\circ$ .

STEP 5

For problem 9(c), the angles are $x^\circ, (x+20)^\circ, (x-50)^\circ$ . The sum of angles in a triangle is $180^\circ$ .
$x + (x+20) + (x-50) = 180$ $3x - 30 = 180$ $3x = 210$ $x = 70$
The angles are $70^\circ, 90^\circ, 20^\circ$ .

STEP 6

For problem 9(d), the two opposite interior angles are $30^\circ$ and $55^\circ$ . The exterior angle is the sum of these two angles.
$\text{Exterior angle} = 30^\circ + 55^\circ = 85^\circ$

STEP 7

For problem 10(a), in a right-angled triangle, one acute angle is $40^\circ$ . The other acute angle is:
$90^\circ - 40^\circ = 50^\circ$

STEP 8

For problem 10(b), in an equilateral triangle, all angles are $60^\circ$ . Therefore, the remaining angles are also $60^\circ$ .

STEP 9

For problem 10(c), in an isosceles triangle, one base angle is $45^\circ$ . The other base angle is also $45^\circ$ .

STEP 10

For problem 10(d), if one angle of an isosceles triangle is equal to the sum of the remaining angles, then:
Let the equal angles be $x$ and the third angle be $2x$ .
$x + x + 2x = 180$ $4x = 180$ $x = 45$
The angles are $45^\circ, 45^\circ, 90^\circ$ .

STEP 11

For problem 11(a), in $\triangle ABC$ , $\angle A = \angle B = 65^\circ$ . Find $\angle C$ :
$\angle C = 180^\circ - 65^\circ - 65^\circ$ $= 50^\circ$

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

STEP 6

For problem 9(d), the two opposite interior angles are 30∘ 30^\circ 30∘ and 55∘ 55^\circ 55∘. The exterior angle is the sum of these two angles.Exterior angle=30∘+55∘=85∘ \text{Exterior angle} = 30^\circ + 55^\circ = 85^\circ Exterior angle=30∘+55∘=85∘

STEP 7

For problem 10(a), in a right-angled triangle, one acute angle is 40∘ 40^\circ 40∘. The other acute angle is:90∘−40∘=50∘ 90^\circ - 40^\circ = 50^\circ 90∘−40∘=50∘

STEP 8

For problem 10(b), in an equilateral triangle, all angles are 60∘ 60^\circ 60∘. Therefore, the remaining angles are also 60∘ 60^\circ 60∘.

STEP 9

For problem 10(c), in an isosceles triangle, one base angle is 45∘ 45^\circ 45∘. The other base angle is also 45∘ 45^\circ 45∘.

STEP 10

STEP 11

For problem 11(a), in △ABC\triangle ABC△ABC, ∠A=∠B=65∘\angle A = \angle B = 65^\circ∠A=∠B=65∘. Find ∠C\angle C∠C:∠C=180∘−65∘−65∘ \angle C = 180^\circ - 65^\circ - 65^\circ ∠C=180∘−65∘−65∘ =50∘ = 50^\circ =50∘

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For problem 9(d), the two opposite interior angles are $30^\circ$ and $55^\circ$ . The exterior angle is the sum of these two angles.
$\text{Exterior angle} = 30^\circ + 55^\circ = 85^\circ$

For problem 10(a), in a right-angled triangle, one acute angle is $40^\circ$ . The other acute angle is:
$90^\circ - 40^\circ = 50^\circ$

For problem 10(b), in an equilateral triangle, all angles are $60^\circ$ . Therefore, the remaining angles are also $60^\circ$ .

For problem 10(c), in an isosceles triangle, one base angle is $45^\circ$ . The other base angle is also $45^\circ$ .

For problem 11(a), in $\triangle ABC$ , $\angle A = \angle B = 65^\circ$ . Find $\angle C$ :
$\angle C = 180^\circ - 65^\circ - 65^\circ$ $= 50^\circ$