Geometry – SpringBoard 2015 Quarter 2

Hunter SmithESUMS

New Haven Public Schools

List of main topics covered on the Quarterly• Setting up and solving equations• Congruent objects have equal measures• Complementary and supplementary

• Add together two pieces to equal a known total (complementary = 90; supplementary = 180)

• Transformations• Rotations (Positive rotations are CounterClockWise), Reflections, Translations• Preimage and Image, Rigid and Non-rigid (Vocabulary);

• Find one from the other using a rule• Infer a rule from Preimage and Image• State whether a transformation is Rigid or Non-rigid

• Image of point or image of shape

Shapes• Triangles

• Congruence • CPCTC – Corresponding Parts of Congruent Triangles are Congruent• Write a congruence statement

• Exterior angle and their relation to “remote interior angles”• Medians and Altitudes (orthocenter, centroid, etc)

• Quadrilaterals• Types of Quadrilaterals (Square, rectangle, rhombus, parallelogram, trapezoid)

• What angles or sides are congruent? What is parallel? Any relation with the diagonals?• Congruent angles and sides

• Set up equations to solve.• Supplementary Angles

• Set up equations to solve.

And some assorted others.• Midsegment theorem• Perpendicular bisectors• Midpoint• Bisector• Two-Column Proofs• Given (usually first or as needed)• Prove (last)• Definitions• Postulates and Theorems

• Triangle Sum Theorem

Section 1• Setting up and solving equations• Congruent objects have equal measures

Example 1: (a) Find x, (b) Find the measure of each angle)

Two angles ( and are congruent (you are told = given)• and (a) Since the angles are congruent, then their measures are equal

= • Move x terms together (Add 17x both sides), and constant terms too (Add 33 both sides)• Simplify to ……………Clearly (if not, divide both sides by 60)

(b) Since , we can find the measures by definition• ……..by substitution we can replace x with 2

• Similarly, and

• Complementary and supplementary• Add together two pieces to equal a known total Complementary = 90 (right angle); supplementary = 180 (straight angle)Two angles form a linear pair (they make a line/straight angle when combined together)

(a) Set up an equation and solve for x. (b) Use x to find the measure of each angle.

and form a linear pair AND 113 and 35(a) Since the angles form a linear pair, their measures are supplementary (add to 180).

Parentheses added for emphasis• Combine like terms (-2x with -6x and 113 with 35) • Get x term by itself on the left (subtract 148 both sides).• Simplify to ……………Clearly (if not, divide both sides by )

(b) Since , we can find the measures by definition• ……..by substitution we can replace x with

• Similarly, and

Section 2

Section 3• Transformations• Rotations (Positive rotations are CounterClockWise), Reflections, Translations• Preimage and Image, Rigid and Non-rigid (Vocabulary);

• Find one from the other using a rule• Infer a rule from Preimage and Image• State whether a transformation is Rigid or Non-rigid

• Image of point or image of shape

Section 4• Triangles• Congruence

• CPCTC – Corresponding Parts of Congruent Triangles are Congruent• Write a congruence statement

Use the information in the diagram to prove that the two triangles which make up the shape are congruent.

StatementAC=BD and AB=CDCB=CB

ReasonGivenReflexive Property (congruent to itself)SSS

Once we have the congruence relationship, we can use CPCTC to tell what pieces of the two triangles will have the same measurement.Notice that AB matches with DC and AC matches with DB.Continue to match all sides and angles (also called the congruence statement) ;

• Exterior angle and their relation to “remote interior angles”

• Medians and Altitudes (orthocenter, centroid, etc)

Altitudes and orthocenter

Circumcenter (point where all altitudes meet)

•

Incenter

Section 5• Quadrilaterals• Types of Quadrilaterals (Square, rectangle, rhombus, parallelogram,

trapezoid)• What angles or sides are congruent? What is parallel? Any relation with the diagonals?

• Congruent angles and sides• Set up equations to solve.• If , then • If , then by substitution.• Solve for x (subtracted 1x and added 38 to both sides)• Finally (division by 8, justified by division property of equality)• Plug back in to verify RS, ST and double check those answers are the same

What shape(s) has(have) a pair of parallel sides that have different lengths?What shape(s) has(have) four congruent sides?What shape(s) has(have) four right internal angles?

Section 6 – More Quadrilaterals• Supplementary Angles

• Set up equations to solve.• If Angle 1 and Angle 2 are consecutive angles in a parallelogram and and what is the measure of

each angle? •

• Like earlier problems, plug in x to find the measures and • Verify these are correct by noticing the angles add up to 180 (132+48).

• Use the picture to the right and similar reasoning as above to find the missing angle. Given CDAB and ACBD.• Since the quadrilateral is parallelogram, consecutive angles add to

180. Therefore, 100 + (30+x) = 180. Solve. x = 50.• Another approach is to, notice that the two triangle are

congruent (we showed this earlier) and so x is equal to the missing measure in triangle ABC (.

• By the triangle sum theorem, the angle must be 50