Question

!50 POINTS! (3 SIMPLE GEOMETRY QUESTIONS)

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Answer 1

Picture 1

1 and 2 are complementary angles, so 90-41=49. Angle 2 is 49.

Angle 3 is equal to angle 1 so it is 41.

Angle 4 is also equal to angle 3 and angle 1 because angles 1 and 4 are opposite exterior angles. So angle 4 is 41.

Angle 5 is equal to angle 2 so it is 49.

Angles 3, 6, and 4 create a triangle therefore they must add up to 180 degrees. Angle is 41 plus angle 4 which is also 41. 41 times two equals 82. 180 minus 82 is 98 which is the measure of angle 6.

Angle 6 plus its vertical angle is equal to 98 times two. That's 196. The 4 angles in the center add up to 360 degrees because they make a full circle. 360 minus 196 is 164. 164 divided by 2 is 82. Angle 7 is 82.

Picture 2

The two angles we are given measures for are congruent because they are opposite exterior angles. That means we can write out the equation 5x-4=2x+8. We need to subtract/add from both sides of the equation to get x by itself. First, we can subtract 2x from each side to make the new equation 3x-4=8. Then we can add 4 to make 3x=12. 3 divided by 3 is 1. 12 divided by 3 is 4. Therefore x is equal to 4.

Picture 3

Just like in the other picture, the two angles we are given are opposite exterior angles so we can write out an equation the same way. 7x+4=3x+32. We need to get x by itself. First, remove 4 from each side to make the equation 7x=3x+28. Then remove 3x from each side to make the equation 4x=28. 4 divided by 4 is 1. 28 divided by 4 is 7. Therefore x is equal to 7.

Answer 2

For 1st picture:

• m∡2=49°
• m∡3=41°
• m∡4=41°
• m∡5=49°
• m∡6=98°
• m∡7=82°

For 2nd picture:

• x =4

For 3rd picture:

• x =7

Explanation:

Note:

Before answering this, we need to know these properties of rectangle.

• A rectangle is a quadrilateral with four right angles.
• Opposite sides are parallel and equal.
• All angles are right angles.
• The diagonals bisect each other.
• The diagonals are equal in length.
• The perimeter is equal to twice the sum of the length and width.
• The area is equal to the product of the length and width.
• The two triangles formed by the diagonals of a rectangle are congruent.
• The two triangles formed by the diagonals of a rectangle are isosceles.
• The two triangles formed by the diagonals of a rectangle are similar.

For 1st picture:

Given:

m∡1= 41°

Solution:

m∡1+m∡2= 90° Since a rectangle is a quadrilateral with four right angles.

Solving for m∡2

m∡2=90°-m∡1

substituting value:

m∡2=90°-41°

m∡2=49°

again,

m∡3=m∡1 since the corresponding angle of a congruent triangle are equal

therefore, m∡3=41°

again,

m∡4=m∡1 being alternate angle

therefore, m∡4=41°

again,

m∡5=m∡2 since the corresponding angle of a congruent triangle are equal

therefore, m∡5=49°

again,

m∡3+m∡4+m∡6=180°

since sum of interior angle of the triangle is 180°

41°+41°+m∡6=180°

m∡6=180°-41°-41°

m∡6=98°

again

m∡6+m∡7=180° being linear pair

m∡7=180°-m∡6

m∡7=180°-98°

m∡7=82°

`\hrulefill`

For 2nd picture:

(5x-4)°=(2x+8)° being alternate interior angle

5x-4=2x+8

adding both side by 4

5x=2x+8+4

5x=2x+12

subtracting both side by 2x

5x-2x=12

3x=12

dividing both side by 3

x=12/3=4

Therefore, x =4.

`\hrulefill\\`

For 3rd picture:

(7x+4)°=(3x+32)° being alternate interior angle

7x+4=3x+32

subtracting both side by 3x and 4.

7x+4-3x-4=3x+32-3x-4

4x=28

diving both side by 4

x=28/4

x=7

therefore, x =7.