Geometry - Triangles
in an isosceles triangle where the measure of angle A and angle C are base angles and the measure of angle B is the vertex, the measure of angle A=angle C= 1/2(___)
(180-angle B)
Suppose that triangle ABC is similar to DEF and that c is the proportionality constant such that c=AB/DE=BC/EF=AC/DF. What is the relation between the area of ABC and DEF in reference to c?
(area of ABC) =c^2(area of DEF)`
In a 30,60,90 triangle, what is the length of the longer leg in relation to the shorter leg
(longer leg)= √3(shorter leg)
Suppose that triangle ABC is similar to DEF and that c is the proportionality constant such that c=AB/DE=BC/EF=AC/DF. What is the relation between the perimeter of ABC and DEF in reference to c?
(perimeter of ABC) = c(perimeter of DEF
What is the formula for the altitude of an equilateral triangle with side of length S?
(s√3)/2
What is the proportionality constant between congruent angles
1
how many obtuse angles are there in an obtuse triangle?
1
Two triangles have 2 sides that are congruent. What additional information is needed to prove both triangles congruent
1. Whether the angles between these two sides are congruent 2. Whether the third sides are congruent
Right triangle ABC has the measure of angle A = 60 degrees. If the length of the segment opposite of angle A = 5√3, what is the length of the hypotenuse?
10
In triangle ABC, angle a =40 and the exterior angle of angleC =140. What is angle B?
100 degrees
IN triangle ABC angle A=70, angle B =60, and angle C = 50. What are the measures of the exterior angles of each
110, 120, 130 degrees
the side lengths of a right triangle ABC are 8 15 17. If the leg adjacent to angle A =8, what is the sine of angle A
15/17
What is the area of an equilateral triangle with side length 8?
16√3
In the AAS thm of triangle congruence, the third angle of a triangle can be found by finding the sum of the other two angles and subtracting free ___ degrees. From there, the ___ theorem of triangle congruence may be used
180, ASA
In an isosceles triangle where angleA and angle C are base angles and angle B is the vertex, what is angle B is relation to angle A
180-2(angle A)
In an isosceles triangle where angle A and angle C are base angles and angle B is the vertex, what is angle B in relation to angle C
180-2(angle C)
triangle ABC has an area of 10 and is a similar triangle with DEF. If the proportionality constant of AB/DE=2 what is the are of DEF
2.5
If triangle ABC is a 30,60,90 triangle and the longest leg is of length 5, what is the area ?
25√3/6
in isosceles triangle ABC, the length of side a=10 and the base= 8, what is the height?
2√21
What is the side length of an equilateral triangle with an altitude of 3?
2√3
what is the height of an equilateral triangle with an area 4√3?
2√3
in isosceles triangle ABC, the length of side a=7 and the height is 4, what is the length of the base?
2√33
Suppose in right triangle ABC , angle A=60, and the leg opposite of angle A = 3√3. Find the measure of the leg adjacent to angle A
3
If triangle ABC is a 30 60 90 triangle and the hypotenuse=7, what is the length of the shortest leg
3.5
The altitude of an equilateral triangle splits the triangle into two congruent right triangles with angles measuring ___.
30,60,90 degrees
A __, ___, 90 triangle is a special right triangle because it is equal to half of an equilateral triangle
30,60In a 30,60,90 triangle, what is the length of the longer leg in relation to the shorter leg
What is the length of the hypotenuse of a right triangle with sides 15 and 36?
39
if triangle bcd and ugh are similar, db=12 and hf=8, what is the ratio between bc and fg
3:2
What is the proportionality constant between two similar triangles with side lengths of 8,10,12 and of 2,5/2 and 3
4
Right triangle ABC has side lengths 3,4,5 and the length of the leg adjacent to angle A=4 what is the cosine of angle A
4/5
A triangle with angles measuring ____ is special because it is an isosceles right triangle and is equal to half of a square
45,45,90
What is the length of the final leg of a right triangle with hypotenuse 50 and one leg of length 14?
48
there are several common integer groupings for the lengths of right triangle sides. One is 3-4-____ and its multiples, another is ____,12,13 and its multiples, and another is 7-___-25 and its multiples
5,5,24
Suppose in right triangle ABC the lengths of the sides are 5,12,13 and the leg adjacent to angle A is 12. What is the tangent of angle A
5/12
What is the altitude of an equilateral triangle whose side length is 10
5√3
the proportionality constant between BCD and FGH = .5. If the perimeter of DEF=12 what is the perimeter ABC
6
A 30, 60, 90 triangle has a short leg of 3. What are the lengths of all other sides and what is the area?
6 , 3√3, 9√3/2
What is the precise angle measure of each angle in an equilateral triangle?
60 degrees
In an isosceles triangle, angles A and C are congruent. If the measure of angle B = 42 degrees, what is the measure of angle A
69 degrees
What is the length of they hypotenuse of a 45 45 90 triangle with a leg of length 6
6√2
a 45 45 90 triangle with a hypotenuse of 12 has a leg of what length?
6√2
In a isosceles triangle, angles A and C are congruent. If the measure of angle A = 55 degrees, what is the measure of angle B?
70 degrees
What is the area of a 45 45 90 triangle with a leg of length 4
8
What is the hypotenuse of a 45 45 90 triangle with area 32?
8√2
In a right triangle, one of the angles measures _____ degrees
90
Suppose two similar triangles have a proportionality constant of 3;5. What is the ratio os their areas?
9:25
What is the formula for the area (A) of an equilateral triangle with side of length S?
A=(s^2√3)/4
what is the formula for the area (A) of a right triangle with hypotenuse of length c and legs of lengths a and b?
A=1/2(a x b)
what is the formula for finding the area (A) of a triangle?
A=1/2(base) x (height)
What is the area of a 45 45 90 triangle in relation to its legs?
A=1/2(leg)^2
IN a 30,60,90 triangle, what is the formula for the area (A) in relation to the lengths of the sides
A=√3/2(shorter side)^2
Which test used to determine triangle similarity states that if two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar?
AA
name the 3 tests used for determining triangle similarity
AA,SSS,SAS
Which theorem of triangle congruence effectively states that two angles and any sides can determine a triangle
AAS
How does the SSS test for triangle similarity differ for the SSS theorem of triangle congruence?
Both use 3 sides to determine the relatedness of two triangles, but the similarity test requires proportionality between sides while the congruence test requires congruence between sides
triangle ABC is similar to triangle DEF indicates that the triangles are similar and that the corresponding vertices are in order. Thus AB/___=BC/____=____/_____
DE,EF,AC/DF
triangle abc is congruent to triangle def indicates that the triangles are congruent and that the corresponding vertices are in order. Thus AB= ____, BC= ____, and AC=____
DE,EF,DF
True or False: Although the ASS or SSA test is not a congruence test, it can be used to test similarity between 2 triangles
False. ASS/SS is not a similarity test because the test angle must be included between the proportional sides
True or False: Two triangles have 2 sides proportional to each other. This means they are similar
False. More information is needed to prove them similar
True or False: Because two triangles have congruent angles, they are congruent.
False. They are similar, not congruent
True or False: Because two triangles have and angle that is congruent, the triangles are congruent if any two sides are also congruent
False: the sides that are congruent must be the ones that surround the angle
In an isosceles triangle where B is the length of the base, A is the length of the legs,and H is the altitude, H= _____.
H=√((a^2-b^2)/4)
State the SAS test for triangle similarity
IF two sides of one triangle are proportional to two sides of another triangle and the angles between these sides are congruent triangles are similiar
What is the SAS inequality theorem of triangles
If 2 corresponding sides of two triangle are congruent but the angle between them is larger on one than the other, the third side of one is larger than the other
What is the SS inequality theorem of triangles?
If 2 corresponding sides on 2 triangles are congruent but the third side of one is larger than the other, than the included angle on one triangle is larger than the other
What is the SSS similarity theorem of congruence?
If all 3 sides of two triangles are congruent, the triangle are congruent. 3 sides determine a triangle
What is the ASA theorem of triangle congruence
If two angles and the sides between them are congruent, the triangles are congruent. Two angles and the sides between them determine a triangle
what is the SAS theorem of triangle congruence
If two sides and the angle between them are congruent, the 2 triangles are congruent. 2 sides and an included angle determine triangle
_____ triangles have at least 2 congruent sides and two congruent angles opposite those sides
Isosceles
A scalene triangle is intersected by a line that is parallel to the base. What can be said about the new triangle formed with the intersecting line as base?
It is similar to the original triangle
if the base and hypotenuse of 2 triangles and the angle between them are congruent, what theorem can be used to prove the congruency of the triangles?
SAS
one triangle has side lengths of 4,6 and 7, while another also has side lengths of 4,6, and 7. what theory can be used to prove their congruence
SSS
In a triangle, what is true about the opposite sides acting two equal angles?
The length of the opposite sides is also equal
If triangle bcd and triangle fgh are similar,what can be said about the measure of angle c and g?
They are congruent
Suppose two right triangles ABC and DEF are smilier. What can be said about the relationships between the tangent of angle c and the tangent of angle F
They are equal
Descriebt he relationship between two triangles, one with sides measuring 21,15, and 9 and the other measuring 14,10,and 6
They are similar
Two triangles have 2 angles that are congruent. What additionala information is needed to prove both triangles congruent
Whether any corresponding side length is congruent between the 2 triangles
two triangles have 2 angles that are congruent. what additional information is needed to prove both triangles are congruent?
Whether any corresponding side length is congruent between the two triangles.
If 2 triangles have 2 sides that are proportional, what additional information is needed to prove them similar
Whether the measure of the angle between them is congruent or whether the third sides are congruent
what is the pythagorean theorem?
a^2 + b^2 = c^2
A triangle has side lengths 6, 6, 15 and angles measuring 70, 70, and 40 degrees. What is its triangle type?
acute isosceles triangle
The formula for the cosine of angle A in right triangle ABC equals the length of the leg ____ to angle A divided by the length of the hypotenuse
adjacent
In a right triangle ABC with right angle C, angle A equals what in relation to angle B?
angle A=90 degrees - angle B
Triangle ABC is similiar to triangle DEF. Thus thus the measure of angle A= ____, the measure of angle B= ____, and the measure of angle C= ____,
angle D, angle E,angle F
Triangles are similar when they are the same shape; corresponding _____ are congruent while corresponding ___ are merely proportional
angles, sides
In an isosceles triangle where B is the length of the base, A is the length of the legs, and H is the altitude, B = _____.
b=2√a^2-h^2
The altitude of a triangle is a line segment drawn from one ___ to the line that contains the opposite side. Altitudes are most often drawn to the ____ of the triangle
base
Usually the ____ of a triangle is the side that is oriented horizontally
base
With two triangles that have the same height, the ratio of their areas equals the ratio of their ______.
bases
In a right triangle, the two acute angles are ____ and their degree measure adds up to ____ degrees.
complementary, 90
the two triangles adjacent to the hypotenuse in a right triangle are ____ ; their sum equals ___ degrees
complementary, 90
in a 45 45 90 triangle, the length of the legs are ___
equal
all equilateral triangles are also ___ triangles
equiangular
What type of triangle has 3 equal side lengths and angle measures?
equilateral
____ triangles have 3 congruent sides and three congruent angles
equilateral
true or false: the exterior theorem of triangles states that for any given vertex, the exterior angle will be greater than the sum if the two remote interior angles.
false: the exterior angle theorem of triangles states that the sum of the remote interior angles equals the exterior angle of the third vertex
The exterior angle inequality theorem of triangles states that the measure of the exterior angle of any vertex will be ____ the measure of either remote interior angle
greater than
The length of the altitude on a triangle is also known as the ____
height
In a triangle, the side opposite the right angle is called the _____
hypotenuse
The formula for the sine of angle A in right triangle ABC equals the length of the leg opposite angle A divided by ____.
hypotenuse
the midpoint of the ___ of a right triangle is equidistant from al vertices on that triangle
hypotenuse
two right triangles can be proved congruent if the ___ and a ___ are congruent
hypotenuse , leg
according to the pythagorean theorem, the length of the ___ squared equals the sum of the squares of the lengths of the ____.
hypotenuse, legs
in a 30 60 90 triangle, what is the length of the hypotenuse in relation to the legs of the triangle?
hypotenuse= 2 (shorter leg)
What is the length of the hypotenuse in a 45,45,90 triangle in relation to its legs
hypotenuse=√2(leg)
What type of triangle has two sides of equal length?
isosceles
A triangle has angle measures of 45, 45, and 90 degrees. What is its triangle type
isosceles right triangle
in a right triangle, the two sides opposite the acute angles are called the ____
legs
in an isosceles triangle, the equal sides are called the ____ and the third side is called the _____.
legs, base
in a right traingle, what is the relation of the lengths of the lengths of the median m and the hypotenuse h
m=1/2h
A triangle has side length 3,6, and 8 and angles measuring 113 degrees, 34 degrees, and 33 degrees. What is its triangle type?
obtuse scalene triangle
If a line ____ to one side of a triangle intersects with other ides, that line sections off a similar triangle to the original one
parallel
Altitudes and medians in similar triangles are ___ to the sides
proportional
the altitude to the hypotenuse of a right triangle splits the triangle into two smaller ____ triangles, each similar to each other and to the original triangle
right
in an isosceles triangle, the altitude splits the triangle into 2 congruent triangle of what kind?
right triangles
in triangles there can be no more than ___ or ____ angle.
right, obtuse
What type of triangle has 3 different side lengths?
scalene
Scalene triangles have 3 different _____ and three different ____
side lengths, angle measurements
The ASS or SSA triangle them is not a congruence test because two ___ and an ____ outside them do not determine a triangle
sides, angle
all equilateral triangles are ____
similar - they have the same shape
the side lengths of a right triangle ABC are 8 15 17. If the leg adjacent to angle C =15, what are the sine, cosine, and tangent of angle C?
sin=8/17; cos= 15/17; tan= 8/15
A line intersects a triangle through two sides. The new, smaller triangle created by the intersecting line is similar to the original triangle if and only if
the intersecting line is parallel to the third line of the original triangle
in a scalene triangle, the longest side is always opposite which angle?
the largest
In the right triangle abc the tangent of angle a is the ratio of the length of the leg opposite to angle a divided by ____.
the length of the leg adjacent to angle a
Two triangles have a congruent angle. what other information is necessary to prove the two similar
the lengths of the sides that surround the angle are proportional or the measure of any other corresponding angles are congruent
define a median
the line drawn from a vertex to the midpoint of the opposite side
triangles ABC and DEF are congruent and have corresponding vertices that are in order. thus the measure of angle A= ____, the measure of angle B= ____, and the measure of angle C= ____,
the measure of angle A=angle D, the measure of angle B = angle E the measure of angle C = angle F
what is the triangle inequality postulate?
the sum of the elgnths of any two sides of a triangle is always greater than the length of the third side
if triangle bcd and fgh are similar and have the same side lengths, what can be said about the relationship between the two triangles?
they are congruent
if triangle bcd and fgh are congruent and ijk is similar to bcd, what can be said about the relationship between angle g and j
they are congruent angles
If triangle bcd and ugh are similar, what can be said about the lengths of cd and gh?
they are proportional
every triangle has six attributes; three angles and three sides. To establish congruence, it is necessary to prove equivalence in ____ attributes(one of which must be a side)
three
how many acute angles are there in an acute triangle
three
When are triangles congruent?
triangles are congruent when they have the same shape and size; that is, when corresponding angles and sides are all congruent
A median divides a triangle into ___ smaller triangles of ___ area
two, equal
in an isosceles triangle, the altitude bisects both the ___ and the ____, hitting the base at its ____.
vertex angle, base, midpoint
in isosceles triangle ABC, if the length of side a=5 and side b=4, what is the height?
√21
In an isosceles triangle where b is the length of the base, A is the length of the legs, and H is the altitude, A=____.
√b^2/4+h^2
