geometry toolbox advanced proofs (3)
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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