2017-2018 Common Core Geometry Midterm Study Guides

Unit 1: Constructions

Unit 2: Unknown Angles

Definition Diagram If two parallel lines are intersected by a transversal, then corresponding angles are congruent

If two parallel lines are intersected by a transversal, then alternate interior angles are equal.

If two parallel lines are intersected by a transversal, then alternate exterior angles are equal.

If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.

An angle bisector divides an angle in half. !∡!"# = !∡!"#

A perpendicular bisector divides a segment in half. • The perpendicular bisector (!"!!!!) goes through the midpoint (m) of

line segment (!"!!!!) • !"!!!!! ≅ !"!!!!!

N

M

L

Angles in a triangle

sum to !"#°

Right Angle An angle whose measure is 90o.

Angle Bisector A line, ray or segment which cuts an angle into two equal parts.

Perpendicular A line is perpendicular to another if it meets or crosses it at right angles.

Segment Bisector A line, ray or segment which cuts another line segment into two equal parts.

Equidistant A point P is equidistant from others if it is the same distance from them.

Linear Pair Two angles that are adjacent (share a leg) and supplementary (add up to 180°)

Parallel Lines Lines are parallel if they lie in the same plane, and are the

same distance apart over their entire length. (Never

intersect)

Isosceles Triangle A triangle which has at least two of its sides equal in length

Vertical Angles

A pair of non-adjacent angles formed by the

intersection of two straight lines

Equilateral Triangle A triangle which has all three

of its sides equal in length.

Complementary Angles: Two angles that add to 90o

Supplementary Angles: Two angles that add to 180o

Adjacent Angles: Two angles that share a common vertex and common side.

                                           

C

DA B

C

DA B

A

B

C

D

J

E

F

H

G

ao bo

Exterior Angle Theorem: The sum of the two non-adjacent interior angles is equal to the sum of the exterior angle.

Unit 3: Rigid Motions – Transformations

A rigid motion preserves side length (distance) and angle measurement

Reflection Rotations Translation You need to state the line of reflection.

Opposite Isometry- Orientation is NOT preserved

You need to state the center, direction, and degrees for each rotation

Direct Isometry – Orientation is preserved

You need to state the amount each point has been translated “up/down” and “left/right”

Rotations, Reflections, Symmetry

a. When we reflect twice across two lines of reflection that intersect, their intersection can be considered the center of rotation. The pre-image can be rotated around the center of rotation to produce the final image.

b. Every point on one side of the line of symmetry has a corresponding point on the other side of the line of symmetry.

c. Does every figure have a line of symmetry? d. Regular Polygon – all equal sides and all equal angles

i. # of sides = # of lines of symmetry e. Rotational Symmetry f. How can we find the minimum number of degrees a figure must be rotated to map it onto

itself? i. !"#

! where n is the number of sides (ex: 60)

g. The other rotations that will map the figure onto itself are multiples of part f degree measure. Start at 0 and add the minimum number until you reach 360. (ex: 0, 60, 120, 180, 240, 300, 360)

** Rigid Motions: Translations/Rotations/Reflections **

A dilation is not a rigid motion because a dilation doesn’t preserve distance!

Unit 4: Congruency

PROVING TRIANGLES ARE CONGRUENT:

!"! ≅ !"!

!!! ≅ !!!

!!" ≅ !!"

!"! ≅ !"!

!" ≅ !"

*Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

SSA and AA are NOT sufficient statements for proving congruent triangles!!

Picture Given Statement Reason

1.

CDAB ⊥

<!ADB&!BDC are right angles

Perpendicular lines intersect to form right angles

!ADB "!BDC All right <’s are ≅

2.

EF intersects GH at J !EJI "!GJF

Vertical <’s are ≅

3.  

M is the midpoint of AB AM !MB

A midpoint divides a segment into two congruent segments

4.

CD bisects AB at P

P is the midpoint of AB

A bisector divides a segment at it’s midpoint

AP ! PB A midpoint divides a segment into

two congruent segments

5.

BD bisects !ABC !ABD "!CBD An angle bisector divides angle into two congruent angles.

6.

AB is a straight line !ABD is supplementary

to !CDB

Linear Pairs are Supplementary

7.

∆LMN is isosceles LN is the base

ML !MN

In an isosceles triangle legs are congruent.

!MLN "!MNL In an isosceles triangle base angles

are congruent.

8.  

LM !MN

∆LMN is isosceles

If legs are congruent, then the triangle must be isosceles.

!MLN "!MNL In an isosceles triangle base angles are congruent.

C

DA B

J

E

F

H

G

C

DA B

MA B

N

M

L

N

M

L

How to determine the image of a dilated line:

! = !" + !

Since dilations preserve parallelism, the slope (m) always remains the same.

If the center of dilation is on the line, the image is the same as the original line. It remains the same. If the center of dilation is not on the line, the y-intercept, (b), is multiplied by the scale factor!

Unit 5A: Dilations

Unit 5B: Similarity

Similar Triangles have congruent corresponding angles AND corresponding sides that are in proportion!

 

Things to look for when proving triangles similar Vertical Angles Reflexive (shared side) Reflexive (angle)

Remember, when classifying a transformation as a dilation, you need to state the center of dilation and the scale factor!

Dilations preserve angle measurement

Dilations preserve parallelism