Pre Algebra Unit 3

Unit 3: Geometry Measurement

Pre Algebra and Applied Math

Section 1: Rigid Transformations
Notes will not be on the slides they will be in class

Section 2: Effect of Transformation
Notes were integrated into Section 1, we will do study island
to apply this section

Section 3: Similarity and Congruence

Vocabulary

●Collinearity of points: any 3 points are said to be
collinear if a straight line goes through them

●Betweenness of points: any point x is between 2
points A and B, if points AXB are collinear and X
lies between A and B

Rules of Similarity
and Congruence

●The betweeness and collinearity of points are
preserved through rigid transformations

Example 1.

In figure LMNO below, points A and B lie on segment

NO. If the figure LMNO is rotated 90°

counterclockwise about the origin to create figure

L'M'N'O', on which segment will points A' and B' lie?

Example 2.

In figure EFGH below, point K is between points H and
E. If the figure EFGH is reflected across the y-axis to
create figure ABCD, between which two points will the
image of point K lie?

Section 4: Similarity and Congruence

Vocabulary:

●Similarity: Same shape by a consistent
proportion

●Congruence: Same shape and size

Example 1.

Heptagon 1 is congruent to heptagon 2 if heptagon 2
can be created by rotating, reflecting, and/or
translating heptagon 1.

Example 2.

Find a series of transformations that shows that
triangle 1 is congruent to triangle 2.

Example 3.

Hexagon 1 is similar to hexagon 2 if hexagon 2 can be
created by rotating, reflecting, and/or translating and
dilating hexagon 1.

Example 4.

Find a series of transformations that shows that
quadrilateral 1 is similar to quadrilateral 2.

Section 5: Triangle Angles and
Relationships

Rules of a triangle

●A triangle has 180 degrees

Example 1.

∠A=60° and ∠B=55°, what is the measure of ∠C?

Example 2.

Find the missing angle in the image.

Rules of exterior
angles

●An exterior angle is the sum of the two
interior angles that are NOT adjacent to
the exterior angle.

Example 3.

In the figure below ∠A=38 and ∠B= 83, What is the
measure of ∠D?

Example 4.

The exterior angles of triangle FGH are ∠J, ∠K, and
∠L and they are adjacent to ∠F, ∠G, and ∠H
respectively. If m∠F is 61, and m∠L is 124, what is
m∠G?

Section 6: Parallel Lines and
Transversals

Rules!

When a set of two parallel lines are cut by a
transversal pairs of congruent and supplementary
angles are formed.

Vocabulary:

● Parallel Lines: lines that will never intersect

●Transversal: line intersecting a set of lines

●Congruent Angles: Angles with the same
measure

●Supplementary Angles: angles that combine to
make 180°

Diagram

Example 1.

If the measure of ∠C equals 76°, find the measure of
∠G. Use the image from the last slide for reference.

Example 2.

If the measure of ∠C equals 76, find the measure of
∠F.

Example 3.

If the measure of ∠A is 104, find the measure of ∠G.

Example 4.

If the measure of ∠C is 76, find the measure of ∠D.

Example 5.

If the measure of ∠C is 76, find the measure of ∠D.

Example 6.

If the measure of ∠C is 76, find the measure of ∠B.

Section 7: Similar Figures

Vocabulary:

●Angle-Angle Similarity: When two triangles have
two corresponding angles that are congruent
they are similar

●Similar: Same shape with a congruent
proportion

Example 1.

Given that lines m and n are parallel, determine how

triangles SQR and VTU can be shown to be similar.

Example 2.

Triangle ABC is similar to triangle DEF. If m∠C = 120°,

what is the measure of ∠F?

Example 3.

What is the measure of ∠S?

Section 8: Pythagorean Theorem

Formula for
Pythagorean
Theorem

a2+b2=c2

Converse of the
Pythagorean
Theorem

●If the sides of a given triangle correspond to the
pythagorean theorem formula then it’s a right
triangle.

Example 1.

In the triangle above, what is the length of side c? Let c
represent the length of side c.

Example 2.

In the triangle above, what is the length of side b?

Example 3.

Michael exercises in a rectangular pool that is 40 feet
wide and 96 feet long. If he swims the diagonal of the
pool, how far does he swim?

Section 9: Distance

Example 1.

Find the distance between A and B

Example 2.

What is the approximate perimeter of the isosceles

triangle with vertices (-6, 2), (1, 5), and (1, -1)?

Section 10.1: Surface Area

Surface Area
Formulas

●Cube- 6a2

●Rectangular Prism- 2(lb+bh+hl)

●Right Prism- 3(bh)+2(½ bh)

●Right Circular Prism- 2ℼr(r+h)

●Right Pyramid- 4(½ bh)+bh

●Triangular Pyramid- base area + ½ (perimeter x slant)

●Right Circular Cone- ℼr(l+r)

●Sphere- 4ℼr2

Example 1.

Find the surface area of the pyramid below if l = 18

mm, w = 10 mm, and h = 12 mm.

Example 2.

Find the surface area of the square pyramid if the base

length, a, is 6 cm and the height, h, is 4 cm.

Section 10.2: Volume

Volume Formulas

●Cube- a3

●Rectangular Prism- bxlxh

●Right Prism-(hxbxl)/2

●Right Circular Prism- ℼr2h

●Right Pyramid- (lxwxh)/3

●Triangular Pyramid- ⅓ bh

●Right Circular Cone- (ℼr2h)/3

●Sphere Hemisphere-(4/3)ℼr3

Example 1.

Find the volume of the triangular pyramid below. The

base (b) of the triangle measures 6 inches, and the

height (h) measures 4 inches. The height (H) of the

triangular pyramid is 8 inches

Example 2.

Find the volume of the pyramid below if s = 18 in and h

= 12 in.

Example 3.

Find the volume of the cone above if r = 3 cm, h = 4

cm, and s = 5 cm.

Example 4.

Find the volume of the sphere above if r = 4 mm.