Geometry (2) Lesson 7, Geometry (2) Lesson 6, Geometry (2) Lesson 5, Geometry (2) Lesson 4, Geometry (2) Lesson 3, Geometry (2) Lesson 2, Geometry (2) Lesson 1
assume that triangle ABC is congruent to triangle PQR. what congruence statements are correct?
-angle A is congruent to angle P -side AC is congruent to side PR
assume that triangle TUV is congruent to triangle WXY. what congruence statements are correct?
-angle Y is congruent to angle V -angle W is congruent to angle T
assume that triangle GHI is congruent to triangle LMN. what congruence statements are correct?
-angle m is congruent to angle h -angle l is congruent to angle g -side IH is congruent to side NM
if two triangles are congruent then what must be true?
-same shape and size -corresponding angles and sides are congruent
what is the greatest number of obtuse angles that a right triangle can contain?
0
a triangle has two sides of lengths of 5 and 14. what value could the length of the third side be? (10, 2, 5, 8)
10
a triangle has two side lengths of 6 and 9. what value could be the length of the third side (10, 7, 12, 4, 15, 2)
10, 7, 12, 4,
what is the sum of the angle measures no matter what size the triangle is?
180 degrees this is known as the angle sum theorem
a triangle has sides measuring 5 and 8 inches. if x represents the length in inches of the third side, what inequality gives the range of possible values for x?
3 < x < 13
how many sides, vertices, and angles does a triangle have?
3 sides 3 vertices 3 angles
to use any triangle congruence shortcut you need to know _______ pairs of corresponding parts are ______
3; congruent
a triangle has two sides of lengths 8 and 10. what value could the length of the third side be? (4, 18, 7, 20, 8, 10)
4, 7, 8, 10
assume triangle RST is congruent to triangle MNO if MN=6 NO=7 MO=11 what is the length of side RS
6
how many congruent parts would a pair of congruent triangles give?
6 pairs
assume that triangle ABC is congruent to triangle DEF if m<A= 70 deg m<B=47 deg m<C= 63 deg what is the measure of <F
63 degrees
when two triangles have only three congruent angles described in them, what is that postulate/theorem?
AA
Leon drew Triangle ABC and Triangle DEF so that Angle A is congruent to Angle D, Angle B is Congruent to Angle E, AB = 4, and DE = 8. Are Triangle ABC and Triangle DEF similar? If so, identify the similarity postulate or theorem that applies.
Similar-AA
what is the best definition of a scalene triangle?
a triangle in which no two sides are congruent
name the significances with acute, obtuse, and right triangles.
acute- all three angles are acute obtuse- one angle measure is obtuse right- one angle has to be 90 degrees
what is the SAS similarity theorem?
an angle that is congruent to another angle of another triangle lengths of the sides are proportional SIMILAR TRIANGLES
corresponding ______ are congruent (EQUAL) corresponding ______ are proportional
angles; sides
why don't we check through the ASA postulate and AAS theorem?
because they reduce to the AA similarity postulate
how is scale factor calculated?
by the length ratios of corresponding sides to the similar triangle
segments that go through the _________________ remain unchanged. segments that don't will become ____________
center of dilation; stretched
shortcuts to prove _________ SSS, ASA, AAS
congruence
true or false? all ___________ postulates and theorems prove _________
congruence; similairty
you sometimes must use other triangle properties and rules to identify the three ____________ parts needed for a congruence postulate or theorem
congruent
what do similar triangles have?
congruent angles proportional sides and same shape
all _______ triangles are _______
congruent; similar
you can find the _______ parts of two congruent triangles by aligning them perfectly on top of each other.
corresponding
what does CPCTC stand for?
corresponding parts of congruent triangles are congruent
if two angles of a triangle are acute, then the third angle
could be acute, obtuse, or right
how do you see if two similar triangles are congruent?
dilate them both
what is the symmetric property?
every congruence statement can be reversed a=b b=a
what is the reflexive property?
every triangle is congruent to itself a=a
true or false? a triangle with two acute angles must be a right angle.
false
true or false? all isosceles triangles are equilateral?
false
true or false? dilating a triangle changes the angles and side length of the triangle
false
true or false? isosceles triangles may be obtuse or acute but never right
false
true or false? it's possible to build a triangle with side lengths of 5, 5, and 10.
false
true or false? order is very important in triangles
false
true or false? you cannot build a triangle with the lengths of the 3 sides.
false
what is the second step in proving that two triangles are congruent?
figure out if there are any other congruent parts, making sure you have at least 3 congruent parts in total
what is the SSS (side-side-side) postulate
if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
what is the AAS (angle-angle-side) theorem?
if two angles and a non-included side of one triangle are congruent to two angles and a NON-INCLUDED side of another triangle, then the two triangles are congruent.
what is the ASA (angle-side-angle) 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 two triangles are congruent.
what is the third angle theorem?
if two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent
what is the AA similarity postulate?
if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
what is the SAS (side-angle-side) 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 two triangles are congruent.
what is the transitive property?
in a chain of congruence statements, the first triangle is congruent to the last (imagine these are triangles) a=b, b=c, c=d a=d
what is special about sides?
included between the two angles it touches opposite the angle it does not touch
what is special about angles?
included between the two sides it touches opposite the side it doesn't touch
what is a true statement about an exterior angle of a triangle?
it forms a linear pair with one of the interior angles of the triangle
how many obtuse or right angles can there be in a triangle?
no more than 1
what do the terms opposite and included mean?
opposite- across from included- between
when lines never intersect the center of dilation they become
parallel
what is the first step to proving that two triangles are congruent?
recognize any congruent parts that are given
what are the three forms of congruence transformation?
rotation reflection translation
what do similar triangles have in common?
same shape
what are the sides of triangle PQR?
side PQ side QR side PR
shortcuts to prove __________ AA
similairty
Frances drew ABC and DEF so that A D, AB = 4, DE = 8, AC = 6, and DF = 12. Are ABC and DEF similar? If so, identify the similarity postulate or theorem that applies. A)similar SSS B) similar SAS C) similar AA D) cannot be determined??
similar-SAS
only some _________ triangles are __________
similar; congruent
triangles are congruent if they have the same
size and shape
what is the SSS similarity theorem?
the lengths of the corresponding sides of two triangles are proportional then they are SIMILAR
what is the exterior angle theorem?
the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles
what is a corollary of this theorem?
the measure of the exterior angles are always greater than the measure of the remote interior angles
after a congruence transformation, the area of a triangle would be ______________________ it was before
the same as
what is the triangle inequality theorem?
the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
what does every exterior angle have?
they have two remote interior angles that DO NOT TOUCH THE EXTERIOR ANGLES
what do transformations do?
this is an action that does not change the shape or size of the triangle
what does dilating mean?
to enlarge or reduce the shape CHANGING SIZE NOT SHAPE
what term best describes a figure formed by three segments connecting three noncollinear points?
triangle
what is the triangle congruency postulate?
triangles with three pairs of congruent sides will always be congruent
true or false? AAA (angle-angle-angle) does NOT guarantee congruence between two triangles.
true
true or false? In triangle JKL and triangle PQR, if side JK is congruent to side PQ side KL is congruent to side QR and <K is congruent to <Q then JKL must be congruent to triangle PQR
true
true or false? SSA (side-side-angle) does NOT guarantee congruence between two triangles.
true
true or false? a triangle with one obtuse angle must also have two acute angles?
true
true or false? all equilateral triangles are isosceles
true
true or false? if a triangle is congruent, then it is similar
true
true or false? if two triangles are congruent, then they can be moved so that they line up perfectly.
true
true or false? two triangles that have the same side lengths will always be congruent?
true
true or false? with similar triangles, the corresponding angles are equal
true
how many exterior angles are there at each vertex? how many in total?
two exterior angles at each vertex total of 6
what is the SSA (side-side-angle) postulate?
two sides and one NON-INCLUDED angle is the same with another triangle MAY NOT BE CONGRUENT
what is the last step in proving that two triangles are congruent?
use a congruence postulate or theorem (SSA, SSS, etc) to show that the triangles are congruent
what is the AAA (angle-angle-angle) postulate?
when you have all 3 angles on one triangle (same on another triangle) MAY OR MAY NOT BE CONGRUENT
what is the similarity notation?
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