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Congruent Triangles

In mathematics, congruent triangles are similar triangles with one of the sides equal in length. In other words, the corresponding sides are proportional to each other. Congruent triangles are commonly used to illustrate proportional relationships and analysis of such relationships. The equal sides of a congruent triangle are always congruent in length, and the angles opposite the equal sides are equal in measure. Thus, the three properties of congruent triangles can be used to prove that two triangles are congruent. In particular, if two triangles have a side and angle pair that is congruent, then the triangles are congruent. The term "congruent" can also be used to refer to pairs of triangles that are similar, meaning that they are similar in size and shape. In this usage, "congruent triangles" are pairs of triangles that are similar in size and shape. The following properties of congruent triangles are all consequences of their corresponding triangle being similar to itself. In addition, the following properties of congruent triangles are all consequences of their corresponding triangle being similar to another triangle. Two triangles are congruent if and only if they have the same size and shape in both the "left" and the "right" triangle. This is because the left and right sides of one triangle are proportional to the left and right sides of the other triangle: the longer the side of the former triangle, the longer the side of the latter triangle by the same factor. Two triangles are congruent if and only if they have the same shape in both the "right" and the "left" triangle. This is because the hypotenuse of one triangle is proportional to the hypotenuse of the other triangle: the hypotenuse of the first triangle is the hypotenuse of the second triangle by the same factor. Two triangles are congruent if and only if they have the same size in both the "right" and the "left" triangle. This is because the length of the third side is proportional to the length of the fourth side in both triangles by the same factor. Two triangles are congruent if and only if they have the same perimeters in both the "left" and the "right" triangle. This is because the length of one of the sides is proportional to the length of the other of the sides in both triangles by the same factor. Two triangles are congruent if and only if they have the same area in both the "right" and the "left" triangle. This is because the area of the base is proportional to the area of the triangle by the same factor. Two triangles are congruent if and only if their corresponding angles are congruent. This is because if the corresponding angles of two triangles are congruent, then they are equal in measure. Two triangles are congruent if and only if their corresponding angles are supplementary. This is because if the corresponding angles of two triangles are supplementary, then they add to 180 degrees. Two triangles are congruent if and only if their corresponding sides are proportional. This is because the length of one of the sides is proportional to the length of the other of the sides in both triangles by the same factor. Two triangles are congruent if and only if their corresponding sides are proportional to the corresponding sides of another triangle. This is because the length of one of the sides is proportional to the length of the other of the sides in both triangles by the same factor. Two triangles are congruent if and only if their corresponding altitudes are proportional. This is because the length of one of the sides is proportional to the length of the other of the sides in both triangles by the same factor. Two triangles are congruent if and only if their corresponding bisectors are proportional. This is because the length of one of the sides is proportional to the length of the other of the sides in both triangles by the same factor. Two triangles are congruent if and only if their corresponding medians are proportional. This is because the length of one of the sides is proportional to the length of the other of the sides in both triangles by the same factor. Two triangles are congruent if and only if the corresponding angles are congruent. This is because the corresponding angles are equal in measure. Two triangles are congruent if and only if the corresponding sides are proportional. This is because the lengths of the corresponding sides are proportional to each other. There are a number of ways of proving that two triangles are congruent. Two triangles are congruent if and only if one of their corresponding angles is congruent. This is because the corresponding angles of two triangles are congruent if and only if

Parts of Congruent Triangles

88 Practice Problems
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02:11
Geometry for Enjoyment and Challenge

Problem Set B
Given: $\underset{A E}{~} \underset{D E}{\operatorname{AEB}} \angle D E C$$\angle A \equiv \angle D$
Conclusion: $\overline{\mathrm{AC}} \approx \overline{\mathrm{BD}}$
(FIGURE CAN'T COPY)

Congruent Triangles
Beyond CPCTC
Jennifer Stoner
01:33
Geometry for Enjoyment and Challenge

For the following figures, identify $\overline{\mathrm{AD}}$ as a median, an altitude, neither, or both according to what can be proved.
A.(FIGURE CAN'T COPY)
B.(FIGURE CAN'T COPY)
C.(FIGURE CAN'T COPY)
D.(FIGURE CAN'T COPY)

Congruent Triangles
Beyond CPCTC
Jennifer Stoner
00:07
Geometry

The equal sides and angles of two congruent triangles can be read from a congruence correspondence between them. Use the fact that $\triangle \mathrm{CAS} \cong \triangle \mathrm{HEW}$ to copy and complete the following equations.
$\mathrm{HW}=$ $____$

Congruent Triangles
Congruent Polygons
Ashley High

SSS, SAS, ASA, AAS & HL with Congruent Triangles

167 Practice Problems
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02:48
Precalculus

In calculus, some applications of the derivative require the solution of triangles. Solve each triangle using the Law of Sines.
In an oblique triangle $A B C, b=30 \mathrm{cm}, c=45 \mathrm{cm},$ and $\gamma=35^{\circ} .$ Find the length of $a .$ Round your answer to the nearest unit.

Trigonometric Functions of Angles
The Law of Sines
Sarah Wharton
01:59
Precalculus

Let $A, B$, and $C$ be the lengths of the three sides with $X, Y$, and $Z$ as the opposite corresponding angles. Write a program to solve the given triangle with a calculator.
$$A=25.7, C=12.2, \text { and } X=65^{\circ}$$

Trigonometric Functions of Angles
The Law of Sines
Sarah Wharton
00:21
Precalculus

Determine whether each statment is true or false.
The Law of Sines applies only to right triangles.

Trigonometric Functions of Angles
The Law of Sines
Sarah Wharton

Congruent Triangle Proofs

115 Practice Problems
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01:26
Understanding Elementary Algebra with Geometry

Determine what additional information is necessary to show that the triangles are congruent by the given theorems.
(FIGURE CANNOT COPY)
$\triangle A B E \cong \triangle C B D$
(a) By SAS
(b) By ASA

Geometry
Congruence
Julie Silva
00:40
Understanding Elementary Algebra with Geometry

Explain why the two triangles are congruent.
(FIGURE CANNOT COPY)

Geometry
Congruence
Julie Silva
00:21
Understanding Elementary Algebra with Geometry

Explain why the two triangles are congruent.
(FIGURE CANNOT COPY)

Geometry
Congruence
Julie Silva

Congruent Triangle Proofs with CPCTC

32 Practice Problems
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03:06
Understanding Elementary Algebra with Geometry

In Exercises $27-34,$ find the length of the missing sides of the given right triangles.
(TRIANGLE CAN'T COPY).

Geometry
Similarity
Carrie Bain
01:58
Understanding Elementary Algebra with Geometry

In Exercises $21-26,$ round your answer to the nearest tenth where necessary.
The corresponding sides of two similar triangles are in the ratio of 4 to $7 .$ If a side of the smaller triangle is $5.8 \mathrm{cm},$ find the length of the corresponding side of the larger triangle.

Geometry
Similarity
Carrie Bain
02:11
Understanding Elementary Algebra with Geometry

In Exercises $11-14,$ find the length of the indicated side.
Find $|\overline{A C}|$
(IMAGE CAN'T COPY).

Geometry
Similarity
Carrie Bain

Congruent Polygons

50 Practice Problems
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01:32
Geometry for Enjoyment and Challenge

The sum of a polygon's angle measures is nine times the measure of an exterior angle of a regular hexagon. What is the polygon's name?

Polygons
Regular Polygons
Jennifer Stoner
02:34
Geometry for Enjoyment and Challenge

Find the measure of an exterior angle of each of the following equiangular polygons.
a A triangle
b A quadrilateral
c An octagon
d A pentadecagon
e A 23 -gon

Polygons
Regular Polygons
Jennifer Stoner
02:06
Geometry for Enjoyment and Challenge

a How many diagonals does a triangle have?
b How many diagonals does a quadrilateral have?
c How many diagonals does a five-sided polygon have?
d How many diagonals does a six-sided polygon have?
e How many diagonals meet at one vertex of a polygon with $n$ sides?
f How many vertices does an n-sided polygon have?
g How many diagonals does an $n$ -sided polygon have?

Parallel Lines and Related Figures
Four-Sided Polygons
Debasish Das

Using Congruent Triangles

46 Practice Problems
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06:23
Geometry for Enjoyment and Challenge

Given: $\mathrm{m} \angle \mathrm{P}+\mathrm{m} \angle \mathrm{R}<180$
$$
\mathrm{PQ}<\mathrm{QR}
$$
Write an inequality to describe the restrictions on $x$
(GRAPH CAN'T COPY)

Congruent Triangles
Angle-Side Theorems
Md. Tajrian Taher
04:31
Geometry for Enjoyment and Challenge

Given: $\angle 3 \approx \angle 6$
$\angle 3$ is comp. to $\angle 4$ $\angle 6$ is comp. to $\angle 5$ Prove: $\triangle \mathrm{EBC}$ is isosceles.
(GRAPH CAN'T COPY)

Congruent Triangles
Angle-Side Theorems
Md. Tajrian Taher
00:45
Geometry

Give the missing statements and reasons in this proof of Transformation Theorem 4. (FIGURE CAN'T COPY)
Theorem. A triangle and its image under an isometry are congruent.
Given: $\triangle \mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$ is the image of $\triangle \mathrm{ABC}$ under a certain isometry.
Prove: $\triangle \mathrm{ABC} \cong \triangle \mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$.
Proof. (TABLE CAN'T COPY)

Transformations
Properties of Isometries
Ian Shi

Coordinate Proofs

46 Practice Problems
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01:20
Glencoe Geometry

Use the following information. The Leaning Tower of Pisa is approximately 60 meters tall, from base to belfry. The tower leans about $5.5^{\circ}$ so the top right corner is 4.5 meters to the right of the bottom right corner.
Position and label the tower on a coordinate plane.

Quadrilaterals
Coordinate Proof with Quadrilaterals
Ashley High
01:08
Glencoe Geometry

Name the missing coordinates for each parallelogram or trapezoid.
CAN'T COPY THE FIGURE

Quadrilaterals
Coordinate Proof with Quadrilaterals
Ashley High
01:31
Glencoe Geometry

The state of Tennessee can be separated into two shapes that resemble quadrilaterals. Write a coordinate proof to prove that $D E F G$ is a trapezoid. All measures are approximate and given in kilometers.
CAN'T COPY THE FIGURE

Quadrilaterals
Coordinate Proof with Quadrilaterals
Ashley High

Equilateral and Isosceles Triangles

94 Practice Problems
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02:37
Geometry for Enjoyment and Challenge

In $\triangle \mathrm{RST}, \mathrm{RS}=\mathrm{x}+7, \mathrm{RT}=3 \mathrm{x}+5,$ and
$\mathrm{ST}=9-x .$ If $\triangle \mathrm{RST}$ is isosceles, is it also equilateral?
(FIGURE CANNOT COPY).

Congruent Triangles
Types of Triangles
Chris Wojturski
03:17
Geometry for Enjoyment and Challenge

draw your own diagram and write "Given:" and "Prove:" statements in terms of your diagram.
Given: An isosceles triangle and the median to the base Prove: The median is the perpendicular bisector of the base. (This sentence contains two conclusions - "the median is perpendicular to the base" and "the median bisects the base."

Lines in the Plaine
The Case of the Missing Diagram
Debasish Das
02:56
Geometry for Enjoyment and Challenge

If the perimeter of $\triangle \mathrm{EFG}$ is $32,$ is $\triangle \mathrm{EFG}$ scalene, isosceles, or equilateral?
(FIGURE CANNOT COPY).

Congruent Triangles
Types of Triangles
Chris Wojturski

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