**Inequalities in Two Triangles Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on inequalities in two triangles.

Before look at the worksheet, if you would like to know the stuff related to inequalities in two triangles,

**Problem 1 :**

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then prove that the included angle of the first triangle is larger than the included angle of the second triangle.

**Problem 2 :**

In triangles ABC and DEF, we have

AB ≅ DE

BC ≅ EF

AC = 12 inches

m∠B = 36°

m∠E = 80°

Which of the following is a possible length for DF ?

8 inches, 10 inches, 12 inches, 23 inches

**Problem 3 : **

In triangles RST and XYZ, we have

RT ≅ XZ

ST ≅ YZ

RS = 3.7 centimeters

XY = 4.5 centimeters

m∠Z = 75°

Which of the following is a possible measure for m∠T ?

60°, 75°, 90°, 105°

**Problem 4 : **

You and a friend are flying separate planes. You leave the airport and fly 120 miles due west. You then change direction and fly W 30° N for 70 miles. (W 30° N indicates a north-west direction that is 30° north of due west.) Your friend leaves the airport and flies 120 miles due east. She then changes direction and flies E 40° S for 70 miles. Each of you has flown 190 miles, but which plane is farther from the airport ?

**Problem 1 :**

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then prove that the included angle of the first triangle is larger than the included angle of the second triangle.

**Solution : **

Let us consider two triangles ABC and DEF such that

AB ≅ DE

BC ≅ EF

AC > DF

**To Prove : **m∠B > m∠E

Begin by assuming that m∠B is not greater than m∠E.

Then, it follows that either m∠B = m∠E or m∠B < m∠E.

**Case 1 : **

If m∠B = m∠E, then m∠B ≅ m∠E.

So, ΔABC ≅ ΔDEF by the SAS Congruence Postulate and AC = DF.

**Case 2 : **

If m∠B < m∠E, then AC < DF by the Hinge theorem.

Both conclusions contradict the given information that

AC > DF

So the original assumption that m∠B is not greater than m∠E cannot be correct.

Therefore,

m∠B > m∠E

**Problem 2 :**

In triangles ABC and DEF, we have

AB ≅ DE

BC ≅ EF

AC = 12 inches

m∠B = 36°

m∠E = 80°

Which of the following is a possible length for DF ?

8 inches, 10 inches, 12 inches, 23 inches

**Solution : **

From the given information, let us draw the two triangles ABC and DEF.

Because the included angle in triangle DEF is larger than the included angle in triangle ABC, the third side DF must be longer than AC.

So, of the four choices, the only possible length for DF is 23 inches. The diagram of the two triangles ABC and DEF above shows that this is possible.

**Problem 3 : **

In triangles RST and XYZ, we have

RT ≅ XZ

ST ≅ YZ

RS = 3.7 centimeters

XY = 4.5 centimeters

m∠Z = 75°

Which of the following is a possible measure for m∠T ?

60°, 75°, 90°, 105°

**Solution :**

Because the third side in triangle RST is shorter than the third side in triangle XYZ, the included angle m∠T must be smaller than m∠Z.

So, of the four choices, the only possible measure for m∠T is 60°.

**Problem 4 : **

You and a friend are flying separate planes. You leave the airport and fly 120 miles due west. You then change direction and fly W 30° N for 70 miles. (W 30° N indicates a north-west direction that is 30° north of due west.) Your friend leaves the airport and flies 120 miles due east. She then changes direction and flies E 40° S for 70 miles. Each of you has flown 190 miles, but which plane is farther from the airport ?

**Solution : **

Begin by drawing a diagram, as shown below. Your flight is represented by triangle PQR and your friend's flight is represented by triangle PST.

Because these two triangles have two sides that are congruent, you can apply the Hinge Theorem to conclude that RP is longer than TP.

So, your plane is farther from the airport than your friend’s plane.

After having gone through the stuff given above, we hope that the students would have understood, inequalities in two triangles.

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