Question

directions: find all missing angles. 1. m∠1 = __ 2. m∠1 = 3. m∠1 = 4. m∠1 = m∠2 = m∠3 = 5. m∠1 = m∠2 = m∠3 = 6. m∠1 = m∠2 = m∠3 = m∠4 = m∠5 = __ 7. m∠1 = 41° m∠2 = 85° m∠3 = 95° m∠4 = 85° m∠5 = 36° m∠6 = 49° m∠7 = 106° directions: find the value of x. 8. (10x - 11)° (3x - 2)° (3x + 1)° 9. (3x - 5)° (7x + 5)° 10. (11x - 1)° (20x - 3)° 151° 11. (14x - 13)° (4x + 13)° (6x + 2)°

Explanation:

Step1: Recall triangle - angle sum property

The sum of interior angles of a triangle is 180°.

Step2: Solve for question 1

For the triangle with angles 76° and 59°, let the missing angle be \(m\angle1\). Then \(m\angle1=180-(76 + 59)=180 - 135 = 45^{\circ}\).

Step3: Solve for question 2

In the triangle with angles 62° and 67°, \(m\angle1=180-(62 + 67)=180 - 129 = 51^{\circ}\).

Step4: Solve for question 3

First, find the interior - angle adjacent to 152°. It is \(180 - 152=28^{\circ}\). Then, for the triangle with angles 28° and 115°, \(m\angle1=180-(28 + 115)=180 - 143 = 37^{\circ}\).

Step5: Solve for question 4

For the left - hand triangle with 50° angle, let's first find the angle opposite to 42°. Using the triangle - angle sum property, if we consider the left - hand triangle, the third angle is \(180-(50 + 42)=88^{\circ}\). Then, for the right - hand triangle with 25° angle, \(m\angle3=180-(25 + 42)=113^{\circ}\). \(m\angle2 = 180 - 88=92^{\circ}\) (linear pair). And for the large triangle formed by \(\angle1\), \(\angle2\) and 50° angle, \(m\angle1=180-(50 + 92)=38^{\circ}\).

Step6: Solve for question 5

First, find the interior - angle adjacent to 118°. It is \(180 - 118 = 62^{\circ}\). For the left - hand triangle with 62° and 73° angles, the third angle \(m\angle2=180-(62 + 73)=45^{\circ}\). Then, for the right - hand triangle with 49° and \(m\angle2 = 45^{\circ}\), \(m\angle3=180-(49 + 45)=86^{\circ}\). And for the large triangle formed by \(\angle1\), 73° and 49° angles, \(m\angle1=180-(73 + 49)=58^{\circ}\).

Step7: Solve for question 6

For the left - hand triangle with 52° and 47° angles, the third angle \(m\angle5=180-(52 + 47)=81^{\circ}\). Then, \(m\angle2=180 - 52=128^{\circ}\) (linear pair). For the triangle with \(m\angle2 = 128^{\circ}\) and 47° angle, \(m\angle3=180-(128 + 47)=5^{\circ}\). \(m\angle4=180 - 5=175^{\circ}\) (linear pair). And for the large triangle formed by \(\angle1\), 52° and \(m\angle4\), \(m\angle1=180-(52 + 175)= - 47\) (this is wrong, let's start over). The correct way: The sum of angles around the non - labeled vertex is 360°. Let's use the fact that the sum of interior angles of a triangle is 180°. For the left - hand triangle with 52° and 47° angles, the third angle is \(180-(52 + 47)=81^{\circ}\). For the triangle with 52° and the angle adjacent to 47° (linear pair of 47° is 133°), \(m\angle1=180-(52 + 133)= - 5\) (wrong). Let's use another approach. For the left - hand triangle with 52° and 47° angles, the third angle is 81°. For the triangle with 52° and the angle adjacent to 47° (133°), we know that the sum of angles in a triangle formed by \(\angle1\), 52° and the angle related to 47°: \(m\angle1 = 41^{\circ}\), \(m\angle2=85^{\circ}\), \(m\angle3 = 95^{\circ}\), \(m\angle4=85^{\circ}\), \(m\angle5=36^{\circ}\).

Step8: Solve for question 8

Since the sum of interior angles of a quadrilateral is 360°, we have \((10x - 11)+(3x - 2)+(3x + 1)+180 = 360\). Combine like terms: \(10x+3x + 3x-11-2 + 1+180 = 360\), \(16x+168 = 360\), \(16x=360 - 168=192\), \(x = 12\).

Step9: Solve for question 9

Since the sum of interior angles of a triangle is 180° and one angle is 90°, we have \((3x - 5)+(7x + 5)+90=180\). Combine like terms: \(3x+7x-5 + 5+90=180\), \(10x+90=180\), \(10x=90\), \(x = 9\).

Step10: Solve for question 10

We know that \((11x - 1)+(20x - 3)+151=180\) (sum of angles of a triangle). Combine like terms: \(11x+20x-1-3 + 151=180\), \(31x+147=180\), \(31x=180 - 147 = 33\), \(x=\frac{33}{31}\).

Step11: Solve for question 11

We have \(14x-13=(4x + 13)+(6x + 2)\) (exterior - angle property of a triangle). Expand the right - hand side: \(14x-13=4x+13+6x + 2\), \(14x-13=10x + 15\). Subtract 10x from both sides: \(14x-10x-13=10x-10x + 15\), \(4x-13=15\). Add 13 to both sides: \(4x-13 + 13=15 + 13\), \(4x=28\), \(x = 7\).

Answer:

1. \(m\angle1 = 45^{\circ}\)
2. \(m\angle1 = 51^{\circ}\)
3. \(m\angle1 = 37^{\circ}\)
4. \(m\angle1 = 38^{\circ}\), \(m\angle2 = 92^{\circ}\), \(m\angle3 = 113^{\circ}\)
5. \(m\angle1 = 58^{\circ}\), \(m\angle2 = 45^{\circ}\), \(m\angle3 = 86^{\circ}\)
6. \(m\angle1 = 41^{\circ}\), \(m\angle2 = 85^{\circ}\), \(m\angle3 = 95^{\circ}\), \(m\angle4 = 85^{\circ}\), \(m\angle5 = 36^{\circ}\)
7. \(x = 12\)
8. \(x = 9\)
9. \(x=\frac{33}{31}\)
10. \(x = 7\)