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=Welcome to Math 20=

You will be asked a series of questions and you will help supply the information needed so that all students will get or gain a better understanding of the unit referred to as "Triangle Congruency"

side side side triangle congruency

__Definition:__ Triangles are __[|congruent]__ when all corresponding sides and interior angles are [|congruent]. The triangles will have the same shape and size, but one may be a mirror image of the other.
 * Congruent triangles** are a special type of similar triangles. Congruent triangles have the same shape (similar triangles) and size.

AAA does not work.
If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not necessarily the same size. For more on this see [|Why AAA doesn't work]. They are called similar triangles (See [|Similar Triangles]).

SSA does not work.
Given two sides and a non-included angle, it is possible to draw two different triangles that satisfy the the values. It is therefore not sufficient to prove congruence. See [|Why SSA doesn't work].

Please note that triangles with three equal [|angles] are **not** necessarily congruent (since the sum of angles is always 180°, regardless of the length of the sides).

**CPCTC**

When two triangles are congruent, all six pairs of corresponding parts (angles and sides) are congruent. This statement is usually simplified as //c//orresponding //p//arts of //c//ongruent //t//riangles are //c//ongruent, or //CPCTC// for short.

Properties of Congruent Triangles
If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved two triangles are congruent, you can find the angles or sides of one of them from the other.

To remember this important idea, some find it helpful to use the acronym [|CPCTC], which stands for "**C**orresponding **P**arts of **C**ongruent **T**riangles are **C**ongruent".

In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.


 * Similar triangles** have the same shape, but the size may be different

Triangles are [|congruent] when all corresponding sides and interior angles are [|congruent]. The triangles will have the same shape and size, but one may be a mirror image of the other One triangle can be a mirror image of the other, but the triangles can still be congruent if the corresponding sides and angles have the same measure. It can be reflected in any direction, up down, left, right or anything in between.


 * 1) In the diagram below, there are two overlapping triangles AQP and BPR. If we know that [[image:http://www.algebralab.org/img/ab919314-4627-4702-8aac-32de6b906f3a.gif width="209" height="21"]], are these triangles congruent?

> Since these triangles share a common side PQ, in this situation, we have **AAS** as in #4. This is because two corresponding angles are equal and a third side PQ, shared by both triangles, must be the same in both triangles. Therefore the other three corresponding parts must also be equal. For example, sides AQ and BP have the same measure. We have



__Side-Angle-Side (SAS) Congruence Postulate__
If two sides ( CA and CB ) and the included angle ( BCA ) of a triangle are congruent to the corresponding two sides ( C'A' and C'B' ) and the included angle ( B'C'A' ) in another triangle, then the two triangles are congruent.


 * two pairs of corresponding angles are congruent (therefore the third pair of corresponding angles are also congruent).
 * the three pairs of corresponding sides are proportional.

Notice the corresponding angles for the two triangles in the applet are the same. The corresponding sides lengths are the same only when the scale factor slider is set at 1.0. Study the side lengths closely and you will find that the corresponding sides are proportional.

==‍Congruent Triangles==


 * Congruent triangles** are a special type of similar triangles. Congruent triangles have the same shape (similar triangles) and size.

Increase/decrease **<A, <B** and **<C** by clicking and dragging the vertices of the left triangle below. Notice the corresponding angles (**<D** **<E** and **<F**) remain congruent. Since the sum of three angles must be 180o, the third pair of corresponding angles must also be congruent when the first two pairs of corresponding angles are congruent.

In the applet above:

<A = <D, <B = <E, <C = <F

Notice the size and shape of the new pairs of triangles remains the same. The patterns you may have observed and need to know for congruent triangles are displayed below. Remember "

≅ " means "is congruent to".

Two triangles are congruent if:


 * all 6 pairs of corresponding angles and sides are congruent.

|| **Corresponding Triangles**

a/f = b/d = c/e = **1** || <B = <E <C = <F || AB = DE BC = EF AC = DF ||
 * **Corresponding Congruent Angles** || **Corresponding Congruent Sides**
 * factor** = **1**
 * Δ ABC ≅ Δ DEF || <A = <D

The **factor** for congruent triangles is 1. Remember this fact for future problem solving activities.


 * If all 3 pairs of corresponding sides in two triangles are the same, then the triangles are congruent.**

Move the corners **A**, **B** or **C** of the triangle above. Watch how Δ DEF changes. Since the 3 pairs of corresponding sides of the two triangles are equal each time you manipulate the applet, the **SSS (Side, Side, Side) Congruence Relation** proves each pair of triangles are congruent.

__Side-Angle-Side (SAS) Congruence Postulate__
If two sides ( CA and CB ) and the included angle ( BCA ) of a triangle are congruent to the corresponding two sides ( C'A' and C'B' ) and the included angle ( B'C'A' ) in another triangle, then the two triangles are congruent.


 * [[image:http://www.analyzemath.com/Geometry/congruence_sas.gif width="402" height="181" align="center" caption="Side-Angle-Side (SAS) Congruence"]] ||
 * Side-Angle-Side (SAS) Congruence ||

**Example 1:** Let ABCD be a parallelogram and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer.


 * [[image:http://www.analyzemath.com/Geometry/congruence_sas_example_1.gif width="252" height="203" align="center" caption="sas Congruence example"]] ||
 * sas Congruence example ||


 * Solution to Example 1:**
 * In a parallelogram, opposite sides are congruent. Hence sides BC and AD are congruent, and also sides AB and CD are congruent.
 * In a parallelogram opposite angles are congruent. Hence angles ABC and CDA are congruent.
 * Two sides and an included angle of triangle ABC are congruent to two corresponding sides and an included angle in triangle CDA. According to the above postulate the two triangles ABC and CDA are congruent.

Congruent Triangles
Triangles that have exactly the same size and shape are called **congruent triangles.** The symbol for congruent is ≅. Two triangles are congruent when the three sides and the three angles of one triangle have the same measurements as three sides and three angles of another triangle. The triangles in Figure [|1] are congruent triangles. 
 * [[image:http://media.wiley.com/Lux/65/18065.nfg023.jpg align="absmiddle" caption="external image 18065.nfg023.jpg"]] ||
 * || **Figure 1** || Congruent triangles. ||  ||
 * [[image:http://media.wiley.com/Lux/65/18065.nfg023.jpg align="absmiddle" caption="external image 18065.nfg023.jpg"]] ||
 * || **Figure 1** || Congruent triangles. ||  ||
 * || **Figure 1** || Congruent triangles. ||  ||
 * || **Figure 1** || Congruent triangles. ||  ||

Congruent Triangles
Three noncolinear points determine a triangle. Draw the three segments connecting the three pairs of points and find three sides, and 3 interior angles, hence the name triangle. Two triangles are congruent if one can be moved on top of the other, so that edges and vertices coincide. The corresponding sides have the same lengths, and corresponding angles are congruent. Assume the edges of one triangle are the same lengths as the corresponding edges of another triangle. Move the first triangle onto the second so that the bases coincide. Do the apexes also coincide? Both apexes are x units away from the left end of the base, and y units from the right end of the base. Since the triangles are oriented the same way, both apexes lie above the base. Draw a circle of radius x centered at the left end of the base, and a circle of radius y centered at the right end of the base. These circles intersect in precisely two points, one above the base and one below. Thus there is only one possible location for the apex. Both apexes coincide, and the first triangle lies directly on top of the second. Corresponding angles coincide, and are congruent. This method of proving triangle congruence is called SSS, for side side side.
 * [[image:http://www.mathreference.com/pic/geo,congru-sss.gif align="center" caption="Side Side Side"]] ||
 * Side Side Side ||

Next assume the angle of a triangle, and the adjacent sides that form that angle, are congruent to the angle and adjacent sides of a second triangle. Move the first triangle onto the second so that side angle side lines up with side angle side. If the left sides line up, and the right sides don't, one angle is inside the other, hence smaller than the other. Since the angles are congruent, both pairs of sides line up. Does the vertex at the end of the left side of the first triangle coincide with the vertex at the end of the left side of the second triangle? Both are a fixed distance from the common apex, along a common line, hence they are the same point. By the same reasoning, the third vertex of the first triangle coincides with the third vertex of the second triangle. The sides and angles all coincide and the triangles are congruent. This method of proving congruence is called SAS, for side angle side.
 * [[image:http://www.mathreference.com/pic/geo,congru-sas.gif align="center" caption="Side Angle Side"]] ||
 * Side Angle Side ||

Assume the base and base angles of two triangles are congruent. Place one on top of the other so that the bases coincide. The apex is now the intersection of two lines, at specific angles to the base. These lines are the same in both triangles, and their intersection is the same point, hence both triangles have the same apex, and coincide. This method of proving congruence is called ASA, for angle side angle.
 * [[image:http://www.mathreference.com/pic/geo,congru-asa.gif align="center" caption="Angle Side Angle"]] ||
 * Angle Side Angle ||

Assume the base, a base angle, and the apex angle of one triangle are congruent to their counterparts in another. Later on we will prove that the angles of a triangle [|add up to 180°]. This means the remaining base angle in the first triangle has the same measure as the remaining base angle in the second. The triangles are congruent by ASA. This method of proving congruence is called SAA, for side angle angle.
 * [[image:http://www.mathreference.com/pic/geo,congru-saa.gif align="center" caption="Side Angle Angle"]] ||
 * Side Angle Angle ||


 * In general, the method of SSA does not guarantee congruence. Place a 5 12 13 right triangle in the corner of a 9 12 15 right triangle, where 12 is the common side. Let the acute base angle be θ. Subtract the first region from the second and find a triangle with sides 13 4 and 15, and angle θ. Now reflect the smaller right triangle through the common side of length 12. This gives a large triangle with sides 13 15 and 14, and angle θ. These are different triangles, yet they have a side side angle congruence. The problem is, we can flip the segment of length 13 back and forth, creating two different triangles. In general, we have congruence when the second side of SSA (not incident to the angle) is at least as long as the first. Then the aforementioned flip is not possible. ||  || || [[image:http://www.mathreference.com/pic/geo,congru-ssa.gif caption="Side Side Angle"]] ||
 * Side Side Angle || ||

Assume the leg and hypotenuse of one triangle are equal to the leg and hypotenuse of another. Like SAA above, this relies on a theorem we haven't proved yet, but I wanted to keep all the triangle correspondence results together. The [|Pythagorean theorem] tells us the remaining legs in the two triangles are equal. Hence they are congruent by SSS. This method of proving congruence is called LH, for leg hypotenuse. ||  Favorites 4666212 41   All Pages 4666214 41
 * A right triangle contains a right angle, whence the other two angles are acute and complementary. The side opposite the right angle is the hypotenuse. The sides adjacent to the right angle are called the legs. We usually orient the triangle so that one of the legs is horizontal, the base, and the other is vertical, the altitude.
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**Congruent triangles** are a special type of similar triangles. Congruent triangles have the same shape (similar triangles) and size. Increase/decrease **<A, <B** and **<C** by clicking and dragging the vertices of the left triangle below. Notice the corresponding angles (**<D** **<E** and **<F**) remain congruent. Since the sum of three angles must be 180o, the third pair of corresponding angles must also be congruent when the first two pairs of corresponding angles are congruent. Two [|triangles] are congruent if their corresponding [|sides] are equal in length and their corresponding [|angles] are equal in size. If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as:
 * [[image:http://upload.wikimedia.org/math/4/f/7/4f7e75d0e804d195e552230de1d3b87f.png caption="triangle mathrm{ABC} cong triangle mathrm{DEF}"]] ||
 * triangle mathrm{ABC} cong triangle mathrm{DEF} ||

In many cases it is sufficient to establish the equality of three corresponding parts and use one of the following results to deduce the congruence of the two triangles. Congruent triangles can be rotated and/or mirror images of each other (reflected). (See [|Congruent triangles] .) In the figure on the right, the two triangles have all three corresponding sides equal in length and so are still congruent, even though one is the mirror image of the other and rotated.