Geometry ToolboxYou will need to use the definitions, postulates, algebraic properties and theorems you have learned to justify your conclusions. Click on the cards below to review each one as needed.

SSS SAS ASA AAS

Triangle Congruency Criteria

reflexive property

Algebraic Properties

Common Definitionsright triangles

congruent

bisector

midpoint

Triangles

Angle Pairs and Parallel Lines

perpendicular linesIsosceles Triangles

parallelograms

rectangles

rhombus

Triangle Angle Sum Theorem

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

alternate Interior angles

corresponding angles

same-side interior anglesQuadrilaterals

vertical angles

complementary angles

supplementary angles

square

Exterior Angles Theoremright angles

Angle Addition Postulate

Postulates

Equilateral Triangles

perpendicular bisector

Line Segments in Trianglesmedians

altitudes

perpendicular bisector

midsegments

angle bisector

isosceles triangles

Properties of parallelograms:

Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent Consecutive angles are supplementary Diagonals are bisect each other

and and

and

and

Opposite sides are parallel Opposite sides are congruent

Opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

Parallelogram

Properties of Rhombuses:

All properties of parallelograms apply to rhombus: Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other

and All sides are congruent Diagonals bisect opposite angles Diagonals are perpendicular

𝑱𝑲 ≅ 𝑲𝑳≅ 𝑳𝑴 ≅𝑴𝑱

𝑲𝑴⊥ 𝑱𝑳

All sides are congruent

Diagonals bisect opposite angles

Diagonals are perpendicular

Rhombus

Properties of Rectangles:

All properties of parallelograms apply to rectangles: Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other

and All angles are congruent Diagonals are congruent 𝑶𝑸≅ 𝑹𝑷

𝑶𝑷⊥𝑷𝑸⊥𝑸𝑹⊥𝑹𝑶All angles are congruent

Diagonals are congruent

Rectangle

Properties of Squares:All properties of parallelograms apply to square: Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other

and* All angles are congruent All sides are congruent Diagonals are congruent Diagonals are perpendicular Diagonals bisect opposite angles*All properties of rectangles and rhombuses applies to squares.

𝑻𝑼⊥𝑼𝑽⊥𝑽𝑾⊥𝑾𝑻𝑻𝑼 ≅𝑼𝑽 ≅𝑽𝑾 ≅𝑾𝑻

𝑻𝑽⊥𝑾𝑼

All angles are congruent All sides are congruent

Diagonals bisect opposite angles

Diagonals are perpendicular

𝑻𝑽 ≅𝑾𝑼Diagonals are

congruent

Square

Complementary angles are two angles whose measures add up to 90. Each angle is called the complement of the other. The angles may or may not be adjacent to each other.

If m HFG=31 and m GFE=59, the ∠ ∠sum is 90. This means that HFG and ∠

GFE are complementary angles. HF ∠is perpendicular to FE. (HF FE)⊥

Complementary Angles

If m IJK=113∠ ° and the m KJL=67∠ °, the sum is 180°. This means that IJK and KJL are ∠ ∠supplementary angles.

IJL is a straight angle.∠

Supplementary angles are two angles whose measures add up to 180°. Each angle is called the supplement of the other. The angles may or may not be adjacent to each other.

Supplementary Angles

Two lines that intersect form four angles. The angles that are opposite from each other are vertical angles.

Line segments MO and NP intersect at point Q and form four angles.

MQN PQO and MQP NQO because ∠ ∠ ∠ ∠vertical angles are congruent.

Vertical Angles

Vertical Angles Theorem:Vertical angles are congruent.

Angle Addition PostulateThe sum of two adjacent angles is equal to the measure of the larger angle that is created.

∠ABC+ CBD= ABD∠ ∠

Angle Addition Postulate

Lines m and n are parallel and are intersected by line t.

There are two pairs of alternate interior angles:

4 6∠ ≅∠3 5∠ ≅∠

Alternate Interior Angles

Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

Alternate interior angles are in between two parallel lines but on opposite sides of the transversal (creates "Z" or backwards "Z")

There are four pairs of corresponding angles:

1 5∠ ≅∠2 6∠ ≅∠4 8∠ ≅∠3 7∠ ≅∠

Corresponding Angles

Corresponding Angles PostulateIf two parallel lines are cut by a transversal, then the corresponding angles are congruent.

Corresponding angles are the angles on the same side of the parallel lines and same side of the transversal.

Lines m and n are parallel and are intersected by line t.

Same-Side Interior Angles Same-Side Interior Angles are the angles between the parallel lines and on the same side of the transversal.

Lines m and n are parallel and are intersected by line t.

Same-Side Interior AnglesIf two parallel lines are cut by a transversal, then same-side interior angles are supplementary.

There are two pairs of same-side interior angles:

4+ 5=180∠ ∠ °3+ 6=180∠ ∠ °

An exterior angle is an angle that is outside of a polygon.

The Triangle Exterior Angle TheoremThe measure of the exterior angle is equal to the sum of the two remote interior angles. The remote interior angles are two interior angles of the triangle that are not adjacent to the exterior angle.

m A + ∠ m B = ∠ m BCD∠

Exterior Angles

Acute Angles of a Right Triangle TheoremIn a right triangle, the two acute angles are complementary.

A right triangle is a triangle with one angle that is 90°. The side opposite the right angle is called the hypotenuse and the two sides that are not the hypotenuse are called legs.

Right Triangles

Therefore, and are complementary angles.

Pythagorean TheoremIn a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

𝐻𝐺2+𝐺𝐹2=𝐻𝐹 2

The Triangle Angle Sum Theorem:The sum of the measures of the angles of any triangle is 180o

Triangle Angle Sum Theorem

The bisector of an angle divides an angle into two congruent angles.

The bisector of a segment divides the segment into two congruent segments (and goes through the midpoint of the segment).

Angle Bisector:EG is a line segment that bisects DGF∠

Line Segment Bisector:LK is a line segment that bisects HJ, point M is the midpoint of HJ

Bisectors

The midpoint of a segment divides a segment into two congruent segments.

Midpoint

If LK is a line segment that bisects HJ, point M is the midpoint of HJ and LK is a line bisector of HJ.

If two triangles share a side, the two sides are congruent.

If two triangles share an angle, the two angles are congruent.

Reflexive Property(shared side or angle)

The reflexive property says that something is congruent to itself

Right angles in triangles create right triangles, so and are right triangles.

Line segments and intersect at point . As shown in the diagram, each angle that is formed is °. ()

From this we can conclude that segment is perpendicular to segment . ()

Perpendicular lines intersect to form 90° angles. (right angles)

Perpendicular Lines

There are six statements that can be written about these triangles based on their corresponding, congruent parts.

∠𝑨≅∠𝑭

∠𝑩≅∠𝑬

∠𝑪≅∠𝑫

𝑩𝑪≅ 𝑫 𝑬𝑨𝑪≅ 𝑫𝑭

𝑨𝑩≅𝑬 𝑭

Six sets of congruent parts!

Corresponding Parts (CPCTC)

Corresponding Parts of Congruent Triangles are Congruent(CPCTC)Corresponding parts can be proved congruent using CPCTC if two triangles have already been proved congruent by one of the triangle congruence criteria (SSS, SAS, ASA, or AAS).

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Since , and, the triangles are congruent.

The congruence statement that relates these two triangles is .

S S S

SSS Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Since , and , the triangles are congruent. (Notice, the angles are in between (included) the two sets of congruent sides.)

The congruence statement that relates these two triangles is .

S S A

SAS Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Since, andthe triangles are congruent. (Notice, the sides are in between (included) the two sets of congruent angles.)

The congruence statement that relates these two triangles is .

A A S

ASA Postulate

If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the triangles are congruent.

Since, andthe triangles are congruent. (Notice, the sides are not between (non-included) the two sets of congruent angles.)

The congruence statement that relates these two triangles is .

A A S

AAS Postulate

Angles, segments or figures that are congruent have exactly the same size and shape. This means that the measures of the angles or lengths of segments are equal.

Since and , we know and are congruent. ()

Congruent

An isosceles triangle is a triangle with two congruent sides.

Example: is an isosceles triangle. Line segments and are congruent and are the legs of . ()Line segment is the base of .and are the base angles of .The vertex of is . The altitude of is . ()

Isosceles Triangles

The base of an isosceles triangle is the side that is not a leg.

The base angles of an isosceles triangle are the angles that are opposite the two legs that are congruent.

The vertex angle is the angle that is not a base angle (the angle that is opposite the base of the isosceles triangle).

The altitude of the isosceles triangle is the line segment that is drawn from the vertex to the base of the isosceles triangle. The altitude of a triangle is always perpendicular to the base

If , then .

Base Angles of Isosceles Triangles TheoremIf a triangle is isosceles, the angles that are opposite the two congruent sides are also congruent.

Altitude of an Isosceles Triangle TheoremIf a line segment is the angle bisector of the vertex angle of an isosceles triangle, then it is also the perpendicular bisector of the base.

In isosceles triangle , is an altitude. CD bisects vertex angle , so . is the perpendicular bisector of , so and bisects .Therefore, is the midpoint of and

Isosceles Triangles

Equilateral triangles have all sides with the same length.

An equiangular triangle is a triangle whose angles all have the same measure.

Equilateral Triangles

A perpendicular bisector is a line segment that divides a segment into two congruent parts and is perpendicular (creates a right angle) with the segment it intersects.

Line segments and intersect at point . Point is a midpoint of since . As shown in the diagram, is perpendicular to because °.

From this we can conclude that segment is a perpendicular bisector of segment .

Perpendicular Bisector

A right angle has a measure of 90°.

∠RST is a right angle. The measure of RST is 90∠ °.

Segment RS is perpendicular to segment ST. (RS ST)⊥

Right Angle

The Midsegment TheoremThe midsegment is parallel to its third side.The midsegment is half of the length of the third side.

Midsegments

The midsegment can be drawn from any two sides of a triangle through the midpoints. The midsegments do not intersect at one point.

The midsegment of a triangle is a segment that joins the midpoints of two sides of a triangle. The midpoint of a segment is the point that divides the segment in half.

The median of a triangle is a segment whose endpoints are a vertex in a triangle and the midpoint of the opposite side.

In this example, the medians intersect at point G. Point G is the centroid of the triangle.

Median

When all three of the medians of a triangle are constructed, the medians of a triangle meet at a point called the centroid.

Another word for centroid is the center of gravity, the point at which a triangular shape will balance.

An altitude of a triangle is a perpendicular segment drawn from a vertex of a triangle to the side opposite. We use the altitude of a triangle when we find the area of a triangle using the formula: where h represents the altitude of the triangle and b represents the base of the triangle (the side that the altitude is drawn to).

If all three altitudes are drawn in a triangle, they meet at a point called the orthocenter.

In this example, the three altitudes of this triangle meet at point R, the orthocenter.

Altitude

The angle bisector is a line segment that divides an angle in half.

The angle bisectors of a triangle intersect at a point called the incenter. The incenter is the center of a circle that can be drawn inside of the triangle (inscribed in the triangle).

The angle bisectors of this triangle intersect at point D, which is the incenter. A circle with center at point D can be inscribed inside ΔUVT.

Angle Bisector

A perpendicular bisector is a line segment that is perpendicular to a line segment and goes through the midpoint of a line segment.

The perpendicular bisectors of the sides of a triangle are concurrent at a point called the circumcenter. This point is the center of a circle that can be circumscribed around the triangle. The red lines represent the perpendicular

bisectors of the sides of ΔFEG. The perpendicular bisectors intersect at point L, the circumcenter. Point L is the center of the circle that is circumscribed around ΔFEG.

Perpendicular Bisectors