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Math 1 Plane Geometry Part 1 Unit Updated July 27, 2016

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Intersecting lines:

When two lines intersect, adjacent angles are supplementary (they make a line and add up to 180

degrees, and vertical angles (angles across from each other are equal).

Parallel lines and transversals:

A line that cuts through two parallel lines is called a transversal. A transversal across parallel lines forms

four equal acute angles and four equal obtuse angles.

Here, line 1 is parallel to line 2. Angles a, c, e, and g are obtuse, so they are all equal. angles b, d, f, and

h are acute, so they are all equal.

Furthermore, any of the acute angles is supplementary to any of the obtuse angles (they make a line and

add up to 180°). Angles a and b are supplementary and so are angles a and d, so are angles a and h, and

so on.

Sample questions:

1. In the figure below, CD is parallel to AB, and PQ intersects CD at R and AB at T. If the measure of

angle CRP = 110°, then what is the measure of angle BTQ?


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2. In the figure to the right. Lines l and m are parallel. Find the measures

of angles, 1, 2, 3, 4, 5, 6, and 7.

Interior angles of a triangle

The three interior angles of any triangle add up to 180°

Sample questions:

3. In triangle DEF at right, the measure of angle EDF is 35 and the

measure of DFE is 65. What is the measure of DEF? (label the

angles)

4. In triangle ABC, angle A is a right angle and angle B measures 30°. What is the measure of angle C?

(Draw a picture and label the angles. Remember that a right angle is 90°).

5. In the following figure, what is the value of X? (remember that supplementary angles = 180°)


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Perimeter and Area of quadrilaterals and other polygons

Perimeter. Perimeter is the outline of a physical area. From Latin, meaning “around” (peri)and

“measure” (metron), a perimeter is basically a boundary of any kind, measuring around the shape. In

mathematics, perimeter refers to the length of this boundary. You might be asked to calculate

the perimeter of a polygon, which is the sum of the length of each side.

Area of a rectangle. From Latin: area - "level ground, an open space,". The number of square units it

takes to completely fill a rectangle. Formula: Width × Height.

Area of a square. The same as for a rectangle, but the width and height are the same so you can also

say 𝑠2 where s is a side of the square.

Area of a parallelogram. width X height or (Base X Height)

Area of a triangle. The formula for area of a triangle is 1

2 of the base times the height.

Area of a trapezoid. A trapezoid can be divided into 2 triangles, then add the two areas.

Sample questions:

6. A homeowner wants to put a wallpaper border on the top edge of all the walls of his kitchen. The

kitchen measures 6.5 meters by 4 meters. What is the required length, in meters, of the border?

7. What is the area, in square units, of the rectangle, shown in the standard (X,Y) coordinate plane

below?


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8. What is the perimeter, in feet, of a rectangle with width 12 feet and length 20 feet?

9. What is the area of parallelogram ABCD?

10. A rectangular rug has an area of 35 square feet, and its width is exactly 2 feet shorter than its

length. What is the length, in feet, of the rug?

11. What is the perimeter, in meters, of the figure to the right?

12. What is the area, in square units, of the triangle shown in the standard (X,Y) coordinate plane

shown?


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13. Andy made a model of the DPC Candy Store which has a side entrance accessed by a ramp. The

dimensions of the ramp on his model are 7 cm high and 24 cm wide. Andy decides to decorate the front

side of the ramp with candy tiles which are 1 cm square. How many candy tiles are needed to cover the

ramp?

14. In triangle XYZ shown, XS and SZ are 3 and 12 units, respectively. If the area of triangle XYZ is 45

square units, how many units long is altitude YS?

Pythagorean theorem

For all right triangles (triangles with a 90° angle), the square of the hypotenuse is equal to the sum of

the squares of the two sides.


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Sample questions:

15. What is the length of the hypotenuse of the triangle below?

16. What is the length of side X in the triangle below?

17. What is the length, in inches, of the diagonal of a rectangle whose dimensions are 16 inches by 30

inches?

18. In the square ABCD shown, the length of AB = 4. What is the length of AC?

19. A zipline is installed from a tower 20 m high crossing a river 50 m wide. What is the length of the zipline wire?


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Circumference of a circle

The perimeter of a circle is called the circumference. It is equal to the diameter times pi.

Area of a circle

The area of a circle is the radius squared multiplied by pi.

Sample questions:

20. The circle in the following figure is inscribed in a square with a perimeter of 24 inches. What is the

area of the shaded region in square inches?

21. Barbara's living room is a rectangle with the dimensions of 14 feet by 12 feet. If the room is

hardwood floor partially covered by a circular throw rug with a diameter of 8 feet, what is the

approximate area of hardwood floor, in square feet, that remains exposed?


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22. In the figure below, the circumference of circle X is 12 𝜋 and the circumference of circle Y is 6 𝜋.

What is the length of XY?

This figure is for questions 23-24. In the following figure, the circle centered at P is tangent to the circle

centered at Q. Point Q is on the circumference of circle P. The length of QP = 6 inches.

23. What is the circumference of circle P?

24. What is the area, in square inches, of circle Q?


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The figure below is for questions 25-27.

In the figure below, O is the center of the circle, C and D are points on the circle, and C, O, and D are

collinear. The length of CD is 16 inches.

25. What is the circumference, in inches, of the circle?

26. What is the length of radius, CO, in inches?

27. What is the area, in square inches, of the circle?

Volume of a rectangular solid

The volume of a rectangular solid is length

times width times height, in other words,

the area of the base times the height.

Volume of a cube is also length X width X

height, but all the dimensions are the same

so it can also be written V = 𝑠3

28. What is the volume, in cubic inches, of the rectangular solid at right?


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29. The recreation center has a swimming pool that is 50 meters long, 25 meters wide, and 2 meters

deep. The pool is surrounded by special non-slip tiles, as shown in the figure below. What is the volume

of water in the pool, in cubic meters, if the pool is only filled half-way?

30. A rectangular back yard pool is 10 meters long, 6 meters wide, and holds 120 cubic meters of water.

If the pool is the same depth in all parts, about how many meters deep is the water in the pool?

More on Parallelograms

A parallelogram has two pairs of parallel sides. Opposite sides are equal. Opposite angles are equal.

consecutive angles (angles adjacent to each other) add up to 180 °.


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31. In the parallelogram shown at the right, angle A = 35°. What are the measures of angles B, C, and D?

32. In parallelogram VWXY shown, points U, V, Y, and Z form a straight line. Given the angle measures

as shown in the figure, what is the measure of angle WVY? What is the measure of angle WXY? What is

the measure of angle XYZ?


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Answers:

1. 110°

2. 1) 135°, 2) 45°, 3)135°, 4) 45°, 5) 45°, 6) 45°, 7) 135°

3. 80°

4. 60°

5. 115°

6. 21m

7. 12 𝑢𝑛𝑖𝑡𝑠2

8. 64 feet

9. 240 𝑢𝑛𝑖𝑡𝑠2

10. 7 feet

11. 82 m

12. 8 𝑢𝑛𝑖𝑡𝑠2

13. 84

14. 6 units

15. 17

16. 6√6 or 14.6969...

17. 34 inches

18. 4√2 or 5.6568...

19. 10√29 or 53.8516...

20. 36 - 9𝜋 or 7.74 𝑖𝑛2

21. 168 - 16𝜋 or 117.76 𝑓𝑡2

22. 9

23. 12𝜋 or 37.68 in

24. 144𝜋 or 452.16 𝑖𝑛2

25. 16𝜋 or 50.24 inches

26. 8 in

27. 64𝜋 or 200.96 𝑖𝑛2

28. 120 𝑖𝑛3

29. 1250 𝑚3

30. 2 m

31. B) 145°, C) 35°, D) 145°

32. angle WVY = 30°, angle WXY = 30°, angle XYZ = 30°

Resources for additional instruction and practice

Glencoe Mathematics Geometry pp. 46, 140, 174-399, 411-414, 520-589. 595-598, 601, 602, 610, 612,

688, 732-733

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