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Questions and Answers
In $\triangle ABC$, if $AB > AC$, which of the following statements must be true according to the Side-Angle Inequality Theorem?
In $\triangle ABC$, if $AB > AC$, which of the following statements must be true according to the Side-Angle Inequality Theorem?
- $m\angle A > m\angle C$
- $m\angle B > m\angle C$
- $m\angle A > m\angle B$
- $m\angle C > m\angle B$ (correct)
Given $\triangle XYZ$ where $XY = 5$, $YZ = 8$, and $ZX = 10$. Which angle has the largest measure?
Given $\triangle XYZ$ where $XY = 5$, $YZ = 8$, and $ZX = 10$. Which angle has the largest measure?
- $\angle Y$
- $\angle Z$
- $\angle X$ (correct)
- Cannot be determined
If in $\triangle PQR$, $m\angle P > m\angle Q$, which of the following statements is true according to the Angle-Side Inequality Theorem?
If in $\triangle PQR$, $m\angle P > m\angle Q$, which of the following statements is true according to the Angle-Side Inequality Theorem?
- $PQ > PR$
- $PQ > QR$
- $PR > QR$
- $QR > PR$ (correct)
In $\triangle DEF$, $DE = 7$, $EF = 5$, and $DF = 9$. Which angle has the smallest measure?
In $\triangle DEF$, $DE = 7$, $EF = 5$, and $DF = 9$. Which angle has the smallest measure?
Which set of side lengths can form a triangle, according to the Triangle Inequality Theorem?
Which set of side lengths can form a triangle, according to the Triangle Inequality Theorem?
Given a triangle with sides of length 7 and 9, what is a possible length for the third side according to the Triangle Inequality Theorem?
Given a triangle with sides of length 7 and 9, what is a possible length for the third side according to the Triangle Inequality Theorem?
According to the Exterior Angle Inequality Theorem, which statement is always true regarding an exterior angle of a triangle?
According to the Exterior Angle Inequality Theorem, which statement is always true regarding an exterior angle of a triangle?
In $\triangle ABC$, if $\angle ACD$ is an exterior angle at vertex $C$, then:
In $\triangle ABC$, if $\angle ACD$ is an exterior angle at vertex $C$, then:
Given two triangles, $\triangle ABC$ and $\triangle DEF$, where $AB = DE$ and $BC = EF$. If $m\angle B > m\angle E$, what can be concluded according to the Hinge Theorem?
Given two triangles, $\triangle ABC$ and $\triangle DEF$, where $AB = DE$ and $BC = EF$. If $m\angle B > m\angle E$, what can be concluded according to the Hinge Theorem?
Two triangles, $\triangle LMN$ and $\triangle PQR$, have $LM = PQ$ and $MN = QR$. If $LN < PR$, which statement is true according to the Converse of the Hinge Theorem?
Two triangles, $\triangle LMN$ and $\triangle PQR$, have $LM = PQ$ and $MN = QR$. If $LN < PR$, which statement is true according to the Converse of the Hinge Theorem?
In $\triangle ABC$, $AB = 6$ and $BC = 8$. Which of the following is NOT a possible length for $AC$?
In $\triangle ABC$, $AB = 6$ and $BC = 8$. Which of the following is NOT a possible length for $AC$?
If $m\angle A = 50^\circ$ and $m\angle B = 60^\circ$ in $\triangle ABC$, which side is the longest?
If $m\angle A = 50^\circ$ and $m\angle B = 60^\circ$ in $\triangle ABC$, which side is the longest?
Given $\triangle PQR$ where $PQ = 4$, $QR = 7$, and $RP = 5$. Arrange the angles from smallest to largest.
Given $\triangle PQR$ where $PQ = 4$, $QR = 7$, and $RP = 5$. Arrange the angles from smallest to largest.
In $\triangle ABC$, if $AB \cong DE$, $BC \cong EF$ and $AC > DF$, what can be said about $\angle B$ and $\angle E$?
In $\triangle ABC$, if $AB \cong DE$, $BC \cong EF$ and $AC > DF$, what can be said about $\angle B$ and $\angle E$?
According to the Triangle Inequality Theorem, what inequality must be true for any $\triangle ABC$?
According to the Triangle Inequality Theorem, what inequality must be true for any $\triangle ABC$?
In $\triangle XYZ$, $\angle X$ is $40^\circ$ and $\angle Y$ is $60^\circ$. Which side of the triangle is the longest?
In $\triangle XYZ$, $\angle X$ is $40^\circ$ and $\angle Y$ is $60^\circ$. Which side of the triangle is the longest?
In triangles $\triangle ABC$ and $\triangle DEF$, $AB = DE$, $BC = EF$, and $\angle ABC > \angle DEF$. Which statement is always true?
In triangles $\triangle ABC$ and $\triangle DEF$, $AB = DE$, $BC = EF$, and $\angle ABC > \angle DEF$. Which statement is always true?
In $\triangle ABC$, if side $a = 5$, side $b = 7$, which of the following is a possible value for side $c$?
In $\triangle ABC$, if side $a = 5$, side $b = 7$, which of the following is a possible value for side $c$?
In triangles $\triangle ABC$ and $\triangle DEF$, $AB = DE$, $BC = EF$. If $AC < DF$, which is true?
In triangles $\triangle ABC$ and $\triangle DEF$, $AB = DE$, $BC = EF$. If $AC < DF$, which is true?
In $\triangle ABC$, $m\angle A = 70^\circ$ and $m\angle B = 50^\circ$. Arrange the sides from shortest to longest.
In $\triangle ABC$, $m\angle A = 70^\circ$ and $m\angle B = 50^\circ$. Arrange the sides from shortest to longest.
Flashcards
Triangle Inequality Theorem 1 (Ss→Aa)
Triangle Inequality Theorem 1 (Ss→Aa)
If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.
Triangle Inequality Theorem 2 (Aa→Ss)
Triangle Inequality Theorem 2 (Aa→Ss)
If one angle of a triangle is larger than a second angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
Triangle Inequality Theorem 3
Triangle Inequality Theorem 3
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Exterior Angle Inequality Theorem
Exterior Angle Inequality Theorem
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Hinge Theorem
Hinge Theorem
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Converse of Hinge Theorem
Converse of Hinge Theorem
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Study Notes
Triangle Inequality Theorem 1 (Ss→Aa) / Side-Angle Inequality Theorem
- If one side of a triangle is longer than a second side, the angle opposite the first side is larger than the angle opposite the second side.
- AM ≅ AI by construction.
- ΔAIM is isosceles based on the definition of an isosceles triangle.
- ∠1 ≅ ∠2 because base angles of an isosceles triangle are congruent.
- ∠1 + ∠3 = ∠AMN according to the Angle Addition Postulate.
- ∠AMN > ∠1 due to the property of equality.
- ∠AMN > ∠2 is derived using the Substitution Property.
- m∠MIN+ m∠N + m∠3 = 180 because the sum of the measures of interior angles of a triangle totals 180.
- m∠2 + m∠MIN = 180° owing to the Linear Pair Theorem.
- m∠MIN + m∠n + m∠3 = m∠2 + m∠MIN through the Transitive Property.
- m∠2 = m∠N+ m∠3 via the Subtraction Property.
- m∠2 > m∠ANM following the property of equality.
- m∠AMN > m∠ANM (from statement 6), due to the Property of Inequality.
Triangle Inequality Theorem 2 (Aa→Ss) / Angle-Side Inequality Theorem
- If one angle of a triangle is larger than a second angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
Triangle Inequality theorem 3
- The sum of the lengths of any two sides of a triangle is greater than the length of the third side
- PL ≅ LE by construction.
- ΔLEP is isosceles according to the definition.
- ∠1 ≅ ∠2 as base angles of an isosceles triangle are congruent.
- ∠LPE ≅ ∠MPE through the Reflexive Property.
- ∠1 ≅ ∠MPE by the Transitive Property.
- m∠MEP = m∠LEM + m∠LEP via the Angle Addition Postulate.
- m∠MEP = m∠LEM + m∠MPE due to the Transitive Property.
- m∠MEP > m∠MPE through the Property of Inequality.
- MP > ME owing to the Property of Inequality.
- MP = PL + LM by the Segment Addition Postulate.
- MP = LE + LM as derived from statements 1 and 10 using substitution.
- LE + LM > ME based on statements 9 and 11 and the substitution property of inequality.
Exterior Angle Inequality Theorem
- The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
- Midpoint Q is on ON, such that OQ ≃ NQ
- Line MR passes through Q, such that MO ≃ OR
- OQ ≃ NQ and MQ ≃ QR by construction.
- ∠3 ≃ ∠4 because they are vertical angles.
- △OQM ≃ △NQR per the SAS Congruence Theorem.
- ∠MON ≃ ∠1 due to CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
- ∠ONP = ∠1 + ∠2 from the Angle Addition Postulate.
- m∠ONP > m∠1 using the Property of Inequality.
- m∠ONP > m∠MON through the Substitution Property.
Hinge Theorem (SAS Triangle Inequality Theorem)
- If two triangles have two congruent sides, the triangle with the larger included angle has the longer third side.
Converse of Hinge Theorem
- If two triangles have two congruent sides, the triangle with the longer third side will have a larger angle opposite that third side
Hinge Theorem or SAS Triangle Inequality Theorem
- If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.
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