3-13 identifying what is needed to prove triangles are congruent
Triangles that have exactly the same size and shape are called congruent triangles. The symbol for coinciding is ≅. Two triangles are congruent when the three sides and the three angles of one triangle have the same measurements every bit 3 sides and three angles of some other triangle. The triangles in Figure 1 are congruent triangles.
Figure 1Congruent triangles.
Corresponding parts
The parts of the two triangles that take the same measurements (congruent) are referred to as corresponding parts. This means that Corresponding Parts of Congruent Triangles are Coinciding (CPCTC). Congruent triangles are named past listing their vertices in respective orders. In Figure , Δ BAT ≅ Δ ICE.
Example 1: If Δ PQR ≅ Δ STU which parts must have equal measurements?
These parts are equal considering respective parts of coinciding triangles are congruent.
Tests for congruence
To show that ii triangles are congruent, it is non necessary to show that all six pairs of respective parts are equal. The following postulates and theorems are the most common methods for proving that triangles are congruent (or equal).
Postulate 13 (SSS Postulate): If each side of one triangle is coinciding to the corresponding side of another triangle, then the triangles are congruent (Figure 2).
Figure 2The respective sides(SSS) of the two triangles are all congruent.
Postulate fourteen (SAS Postulate): If two sides and the angle between them in one triangle are coinciding to the corresponding parts in some other triangle, then the triangles are congruent (Figure three).
Figure iiiTwo sides and the included angle(SAS) of ane triangle are congruent to the
corresponding parts of the other triangle.
Postulate 15 (ASA Postulate): If ii angles and the side between them in i triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Effigy 4).
Effigy 4Two angles and their common side(ASA) in one triangle are congruent to the
corresponding parts of the other triangle.
Theorem 28 (AAS Theorem): If two angles and a side not between them in i triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure v).
Effigy vTwo angles and the side opposite 1 of these angles(AAS) in one triangle
are congruent to the respective parts of the other triangle.
Postulate 16 (HL Postulate): If the hypotenuse and leg of one correct triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 6).
Figure 6 The hypotenuse and one leg(HL) of the kickoff right triangle are congruent to the
respective parts of the second right triangle.
Theorem 29 (HA Theorem): If the hypotenuse and an acute angle of one right triangle are congruent to the corresponding parts of some other correct triangle, then the triangles are congruent (Figure 7).
Figure sevenThe hypotenuse and an acute bending(HA) of the showtime right triangle are congruent
to the corresponding parts of the 2nd right triangle.
Theorem 30 (LL Theorem): If the legs of one right triangle are coinciding to the corresponding parts of another right triangle, and then the triangles are congruent (Figure 8).
Figure viiiThe legs(LL) of the first right triangle are coinciding to the corresponding parts
of the second right triangle.
Theorem 31 (LA Theorem): If one leg and an astute angle of one correct triangle are congruent to the corresponding parts of another right triangle, and then the triangles are congruent (Effigy 9).
Effigy 9One leg and an astute bending(LA) of the first right triangle are congruent to the
corresponding parts of the second right triangle.
Example ii: Based on the markings in Figure 10, complete the congruence statement Δ ABC ≅Δ .
Figure xCoinciding triangles.
Δ YXZ, because A corresponds to Y, B corresponds to Ten, and C corresponds, to Z.
Example 3: Past what method would each of the triangles in Figures eleven (a) through 11 (i) be proven congruent?
Effigy 11 Methods of proving pairs of triangles coinciding.
- (a) SAS.
- (b) None. There is no AAA method.
- (c) HL.
- (d) AAS.
- (e) SSS. The third pair of congruent sides is the side that is shared past the ii triangles.
- (f) SAS or LL.
- (g) LL or SAS.
- (h) HA or AAS.
- (i) None. There is no SSA method.
Example 4: Proper name the additional equal corresponding office(s) needed to prove the triangles in Figures 12 (a) through 12 (f) congruent past the indicated postulate or theorem.
Figure 12Additional information needed to testify pairs of triangles coinciding.
- (a) BC = EF or AB = DE ( but not AC = DF considering these 2 sides lie between the equal angles).
- (b) GI = JL.
- (c) MO = PO and NO = RO.
- (d) TU = WX and SU = VX.
- (due east) m ∠ T = yard ∠ E and chiliad ∠TOW = m ∠ EON.
- (f) IX = EN or SX = TN (merely not IS = ET because they are hypotenuses).
Source: https://www.cliffsnotes.com/study-guides/geometry/triangles/congruent-triangles
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