Learn – Geometry – Triangles – Reals and rationals

1 Congruence

1.1 Exercises

1.1.1 Problem set

1.1.2 Problem set

1.1.3 Problem set

1.1.4 Problem set

2 Angle and side relations

1 Congruence

1.1 Exercises

1.1.1 Problem set

For each of the following problems, assume there are two triangles $ABC$ and $DEF$ , and using the information given, answer with justification whether the two triangles are congruent or not.

$AB = DF, \angle B = \angle F, BC = FE$
$BC = FE, \angle C = \angle E, AC = DE$
$AB = EF, \angle B = \angle D, BC = FD$
$AB = DE, \angle B = \angle E, BC = DF$
$BC = ED, \angle C = \angle D, AC = FE$
$BC = EF, \angle C = \angle F, AB = DE$
$CA = EF, \angle C = \angle E, AB = FD$
$BA = DF, \angle B = \angle D, AC = FE$
$CB = DE, \angle B = \angle E, AB = EF$
$BA = DE, \angle B = \angle E, CB = EF$
$\angle A = \angle F, \angle B = \angle E, \angle C = \angle D$

1.1.2 Problem set

For each of the following problems, assume there are two triangles $ABC$ and $DEF$ , and using the information given, answer with justification whether the two triangles are congruent or not.

$AB = FE, \angle A = \angle F, \angle B = \angle E$
$BC = ED, \angle B = \angle E, \angle C = \angle D$
$AB = FE, \angle A = \angle F, \angle B = \angle D$
$BC = DE, \angle B = \angle D, \angle A = \angle E$
$AC = DF, \angle A = \angle D, \angle B = \angle E$
$AB = FD, BC = DE, CA = EF$

1.1.3 Problem set

In $\Delta ABC$ , $AB = 5, BC = 4$ . Further, $D$ is a point on $BC$ such that $\angle BAD = \angle CAD$ and $AD \perp BC$ . What is the measure of $AC$ ?
In $\Delta PQR$ , $PR = 7, QR = 3$ . Further, $S$ is a point on $QR$ such that $PS \perp QR$ and $QS=RS$ . What is the measure of $PQ$ ?
In $\Delta DEF$ , $\angle D = 70^\circ$ . Further, $G$ is a point on $DF$ such that $EG \perp DF$ and $\angle DEG = \angle GEF$ . What is the measure of $\angle DEF$ ?

1.1.4 Problem set

In $\Delta ABC$ , $AB = 6$ and $AC = 4$ . And, $P$ is the mid-point of side $AB$ so that $AP=PB=3$ . Then, $Q$ is a point on $AC$ such that $PQ || BC$ . What is the measure of $QC$ ?

2 Angle and side relations

2.1 Exercises

2.1.1 Problem set

Prove that in a triangle, if two angles are congruent, the sides opposite to them are also congruent.
Prove that in a triangle, if two sides are congruent, the angles opposite to them are also congruent.
Which of the following are possible in a triangle?
1. $\angle A = 47^\circ, \angle B = 86^\circ, \angle C = 47^\circ, AB = 5, BC = 7$
2. $\angle A = 50^\circ, \angle B = 70^\circ, \angle C = 60^\circ, AB = 3, BC = 4$
3. $\angle A = 65^\circ, \angle B = 45^\circ, \angle C = 70^\circ, AC = 7, AB = 4$
Which of the following are possible lengths of sides of a triangle?
1. $3,5,9$
2. $\frac{2}{9},\frac{3}{4},1$
3. $x, x+1, x+2$ , given that $x > 0$
Two sides of a triangle are $3$ and $5$ . What are the smallest and the largest possible lengths for the third side of a triangle?

3 Miscellaneous

3.1 Exercises

3.1.1 Problem set

In the following figure, $AB \parallel CD$ . If $GE = 5$ , justify that $HF = 5$ .
In the following figure, $\angle ABC$ is a right angle. And, $ACGF$ and $BCED$ are squares.

Area of square $ACGF$ is $169$ square units, and area of square $BCED$ is $144$ square units. What is the area of $\Delta ABC$ ?
Area of an equilateral triangle $ABC$ is $p$ square units. $DEF$ is another equilateral triangle whose side is one-third of the side of $\Delta ABC$ . What is the area of $\Delta DEF$ ?
In the following figure, $BD$ and $CE$ are straight line segments. $AC\parallel BD$ and $AB\parallel CD$ . What is the area of the parallelogram $ABDC$ ?
In the above problem, what is the area of the polygon $AEFDC$ ?
In the following figure $ABDC$ is a parallelogram, and $F$ is the point of intersection of diagonals $BC$ and $AD$ . What is the area of $\Delta BFD$ ?