Geometric TwinsClass 7 Mathematics NCERT Solutions

To Check: If the two given L-shaped figures are congruent.

Method: Congruent figures have the same shape and size. We can check this by comparing their corresponding side lengths and angles.

Figure 1 Description: An L-shaped polygon. Let's list its side lengths in clockwise order starting from the top left corner: a horizontal side of length 2, a vertical side of length 4, a horizontal side of length 4, a vertical side of length 2, a horizontal side of length 6, and a vertical side of length 6. All angles are right angles ($90^\circ$).

Figure 2 Description: A rotated L-shaped polygon. Let's list its side lengths in clockwise order starting from the top right corner: a vertical side of length 2, a horizontal side of length 4, a vertical side of length 4, a horizontal side of length 2, a vertical side of length 6, and a horizontal side of length 6. All angles are right angles ($90^\circ$).

Comparison: Both figures have identical sets of side lengths: {2, 4, 4, 2, 6, 6}. Both figures have all internal angles as right angles. Since all corresponding sides and angles are equal, the two figures have the same shape and size. The second figure can be rotated to perfectly superimpose on the first figure.

Conclusion: Yes, the two figures are congruent.

To identify: Which pairs of figures are congruent.

Final Answer: The pair of keys is the only congruent pair.

(a) Circle: To create a congruent circle, you only need to measure its radius (or diameter). A circle is completely defined by this single measurement.

(b) Rectangle: To create a congruent rectangle, you need to measure its length and its breadth. These two measurements define the size and shape of the rectangle (as all angles are $90^\circ$).

(a) Two Circles: To check if two circles are congruent, you would compare their radii. Two circles are congruent if and only if their radii are equal.

(b) Two Rectangles: To check if two rectangles are congruent, you would compare their dimensions. Two rectangles are congruent if and only if the length and breadth of one rectangle are equal to the corresponding length and breadth of the other rectangle.

Figure Description: The figure is a cross-like shape made of four identical line segments originating from a central point, with equal angles between adjacent segments.

Method: To check if two such figures are congruent, we need to verify two things:

Given: Two L-shaped figures are provided. The second figure appears to be a mirror image of the first figure.

To Check: If the two figures are congruent.

Solution: Congruent figures are those that have the same shape and size. They can be superimposed on each other exactly. This can sometimes involve rotating or flipping one of the figures.

In this case, the first L-shaped figure can be flipped over (reflected) and then placed exactly on top of the second L-shaped figure. Since one figure can be made to fit exactly over the other by flipping it, they have the same shape and size.

Final Answer: Yes, the two figures are congruent.

To Find: The pairs of figures that appear to be congruent.

Solution: We need to visually inspect each pair of figures to see if they have the same shape and size. A congruent pair can be superimposed exactly, possibly after rotation or flipping.

Final Answer: The pairs of keys, leaves, and birds appear congruent.

(a) Circle: To create a congruent circle, you only need one measurement: the radius (or the diameter). All circles with the same radius are congruent.

(b) Rectangle: To create a congruent rectangle, you need two measurements: the length and the breadth. All rectangles with the same length and breadth are congruent.

(a) Circles: To check if two circles are congruent, you would measure their radii (or diameters). If the radii of the two circles are equal, the circles are congruent.

(b) Rectangles: To check if two rectangles are congruent, you would measure their lengths and breadths. If the length of the first rectangle is equal to the length of the second, and the breadth of the first is equal to the breadth of the second, then the two rectangles are congruent.

Given: A figure composed of four line segments forming a shape with four vertices (a kite), and two pairs of these figures.

How to check for congruence: To check if two such figures are congruent, we need to ensure that they have the same shape and size. This can be done by measuring all four corresponding side lengths and the corresponding angles. If all corresponding sides and angles are equal, the figures are congruent. A simpler way, since the figure can be seen as two triangles sharing a common base (the longer diagonal), is to check if the corresponding triangles are congruent. This would mean checking the lengths of the two distinct sides and the angle between them, or all three side lengths of each constituent triangle.

Given: $\triangle \mathrm{HEN} \cong \triangle \mathrm{BIG}$

To Find: All other correct ways of expressing this congruence.

Solution: The given congruence statement implies the following correspondence between vertices: $\mathrm{H} \leftrightarrow \mathrm{B}$ $\mathrm{E} \leftrightarrow \mathrm{I}$ $\mathrm{N} \leftrightarrow \mathrm{G}$

To write other correct congruence statements, we must maintain this correspondence. We can start the naming from any vertex, as long as the order of corresponding vertices is preserved.

The 6 correct ways are:

Final Answer: The other five correct ways are: $\triangle \mathrm{HNE} \cong \triangle \mathrm{BGI}$ $\triangle \mathrm{EHN} \cong \triangle \mathrm{IBG}$ $\triangle \mathrm{ENH} \cong \triangle \mathrm{IGB}$ $\triangle \mathrm{NHE} \cong \triangle \mathrm{GBI}$ $\triangle \mathrm{NEH} \cong \triangle \mathrm{GIB}$

Given: A quadrilateral ABCD with a diagonal AC. It is given that $\mathrm{AB}=\mathrm{AD}$ and $\mathrm{CB}=\mathrm{CD}$.

Yes, we can identify a pair of congruent triangles: $\triangle \mathrm{ABC}$ and $\triangle \mathrm{ADC}$.

Reasoning: We can compare the sides of $\triangle \mathrm{ABC}$ and $\triangle \mathrm{ADC}$.

Since all three corresponding sides of the triangles are equal, the triangles are congruent by the SSS (Side-Side-Side) congruence condition. So, $\triangle \mathrm{ABC} \cong \triangle \mathrm{ADC}$.

Yes, the diagonal AC divides both $\angle \mathrm{BAD}$ and $\angle \mathrm{BCD}$ into two equal parts.

Reasoning: Since we have proved that $\triangle \mathrm{ABC} \cong \triangle \mathrm{ADC}$, their corresponding parts must be equal. This is often referred to as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Final Answer: Yes, $\triangle \mathrm{ABC}$ and $\triangle \mathrm{ADC}$ are congruent by the SSS condition. Yes, AC divides $\angle \mathrm{BAD}$ and $\angle \mathrm{BCD}$ into two equal parts because they are corresponding angles of congruent triangles.

Given: Two triangles, $\triangle \mathrm{DFE}$ and $\triangle \mathrm{DGE}$, which share a common side DE. It is given that $\mathrm{DF}=\mathrm{DG}$ and $\mathrm{FE}=\mathrm{GE}$. Note: The question asks about $\triangle DFE$ and $\triangle GED$. Let's assume the vertex correspondence is D-G, F-E, E-D. This seems unusual. Let's assume the intended comparison is between $\triangle DFE$ and $\triangle DGE$.

Let's compare $\triangle DFE$ and $\triangle DGE$:

Since all three corresponding sides of $\triangle DFE$ and $\triangle DGE$ are equal, the triangles are congruent by the SSS (Side-Side-Side) congruence criterion. So, $\triangle \mathrm{DFE} \cong \triangle \mathrm{DGE}$.

Now, let's consider the congruence as stated in the question: $\triangle \mathrm{DFE}$ and $\triangle \mathrm{GED}$. This implies the correspondence: $\mathrm{D} \leftrightarrow \mathrm{G}$ $\mathrm{F} \leftrightarrow \mathrm{E}$ $\mathrm{E} \leftrightarrow \mathrm{D}$

Let's check the corresponding sides:

Thus, the statement $\triangle \mathrm{DFE} \cong \triangle \mathrm{GED}$ is not necessarily true.

Assuming a typo in the question and the intended comparison was $\triangle DFE$ and $\triangle DGE$.

Final Answer: Assuming the question meant to ask if $\triangle \mathrm{DFE}$ and $\triangle \mathrm{DGE}$ are congruent, then yes, they are congruent by the SSS condition. The correct congruence statement is $\triangle \mathrm{DFE} \cong \triangle \mathrm{DGE}$.

Given: Two pairs of triangles with some side and angle measurements.

Given: A figure with two triangles, $\triangle OAB$ and $\triangle OCD$, formed by intersecting lines AC and BD. It is given that line segment AB is parallel to line segment CD, and $AB = CD$.

To Find: Other equal parts and to check for congruence.

Expressing the congruence: Based on the ASA criterion for $\triangle OAB$ and $\triangle OCD$, the correspondence is $A \leftrightarrow C$, $B \leftrightarrow D$, $O \leftrightarrow O$. So, $\triangle OAB \cong \triangle OCD$. Based on the SAS criterion for $\triangle ABD$ and $\triangle CDB$, the correspondence is $A \leftrightarrow C$, $B \leftrightarrow D$, $D \leftrightarrow B$. So, $\triangle ABD \cong \triangle CDB$.

Final Answer: The other equal parts are $\angle OAB = \angle OCD$, $\angle OBA = \angle ODC$, and $\angle AOB = \angle COD$. Also, as a result of congruence, $OA=OC$ and $OB=OD$. Yes, the two triangles are congruent. The congruence can be expressed as $\triangle OAB \cong \triangle OCD$ (by ASA) or $\triangle ABD \cong \triangle CDB$ (by SAS).

Given: Two triangles, $\triangle ABC$ and $\triangle DBC$, sharing a common side BC. It is given that $\angle ABC = \angle DBC$ and $\angle ACB = \angle DCB$.

To Show: $\angle BAC = \angle BDC$ and determine if the triangles are congruent.

Final Answer: Yes, the two triangles $\triangle ABC$ and $\triangle DBC$ are congruent by the ASA condition. Because the triangles are congruent, their corresponding angles $\angle BAC$ and $\angle BDC$ must be equal.

Given: A quadrilateral ABCD with diagonals AC and BD intersecting. We are given $\angle ABD = \angle DCA$ and $\angle ACB = \angle DBC$.

To Find: The equal parts in the figure.

Solution: Let's consider $\triangle ABC$ and $\triangle DCB$.

Now, let's use the ASA condition on $\triangle ABC$ and $\triangle DCB$.

By ASA congruence condition, $\triangle ABC \cong \triangle DCB$.

Since the triangles are congruent, their corresponding parts are equal (CPCTC). The correspondence is $A \leftrightarrow D$, $B \leftrightarrow C$, $C \leftrightarrow B$.

The equal parts are:

Final Answer: The equal parts are: Sides: $AB = DC$ and $AC = DB$. Angles: $\angle BAC = \angle CDB$ and $\angle ABC = \angle DCB$.

Given: The congruence statement $\triangle \mathrm{AIR} \cong \triangle \mathrm{FLY}$.

To Find: The corresponding vertices, sides, and angles.

Solution: The order of the letters in the congruence statement indicates the correspondence.

To Find: Which pairs of triangles are congruent and the reason.

Given: Two line segments AD and BC intersect at point O such that $OA = OD$ and $OB = OC$.

To Prove: Line segment AB is parallel to line segment CD ($AB \parallel CD$).

Proof: First, we will prove that $\triangle OAB$ is congruent to $\triangle ODC$. Consider $\triangle OAB$ and $\triangle ODC$:

By the SAS (Side-Angle-Side) congruence condition, we can conclude that $\triangle OAB \cong \triangle ODC$.

Since the triangles are congruent, their corresponding parts are equal (CPCTC). Therefore, the corresponding angles are equal. $\angle OAB = \angle ODC$.

Now, consider the lines AB and CD with AD as a transversal line cutting through them. The angles $\angle OAB$ (which is the same as $\angle DAB$) and $\angle ODC$ (which is the same as $\angle ADC$) are a pair of alternate interior angles. Since we have proved that these alternate interior angles are equal, the lines AB and CD must be parallel.

Hence Proved. AB is parallel to CD.

Given: ABCD is a square.

Proof: In a square, all four sides are equal. So, $AB = BC = CD = DA$. Consider $\triangle ABC$ and $\triangle ADC$:

By the SSS (Side-Side-Side) congruence condition, $\triangle ABC \cong \triangle ADC$. Hence Proved.

Yes, $\triangle ABC$ is also congruent to $\triangle CDA$. Proof: Let's check the correspondence for $\triangle ABC \cong \triangle CDA$:

An isosceles triangle provides such an example. Let $\triangle PQR$ be an isosceles triangle with $PQ=PR$. If we reflect it across its altitude from P, we get a congruent triangle $\triangle P'Q'R'$. Let's consider two separate but congruent isosceles triangles, $\triangle PQR$ and $\triangle XYZ$, where $PQ=PR=XY=XZ$. Then we have:

Two congruent equilateral triangles. Let $\triangle ABC$ and $\triangle PQR$ be equilateral triangles with side length 's'. Since all sides are equal ($AB=BC=CA=s$ and $PQ=PR=RQ=s$), any vertex of $\triangle ABC$ can correspond to any vertex of $\triangle PQR$. The total number of ways to map the vertices is $3 \times 2 \times 1 = 6$. The six congruences are:

Given: A triangle ABC where A is the center of a circle and vertices B and C lie on the circle. The angle at the center, $\angle A$, is given as $80^\circ$.

To Find: The measures of $\angle B$ and $\angle C$.

Solution: Since A is the center of the circle and B and C are points on the circle, the line segments AB and AC are both radii of the circle. Therefore, the lengths of these segments are equal: $AB = AC$.

In $\triangle ABC$, since two sides (AB and AC) are equal, it is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are also equal. So, $\angle C$ (opposite side AB) must be equal to $\angle B$ (opposite side AC). $\angle B = \angle C$.

The sum of the angles in any triangle is $180^\circ$. So, in $\triangle ABC$: $\angle A + \angle B + \angle C = 180^\circ$

We are given $\angle A = 80^\circ$. Substituting the values: $80^\circ + \angle B + \angle B = 180^\circ$ (since $\angle C = \angle B$) $80^\circ + 2 \angle B = 180^\circ$ $2 \angle B = 180^\circ - 80^\circ$ $2 \angle B = 100^\circ$ $\angle B = \frac{100^\circ}{2}$ $\angle B = 50^\circ$

Since $\angle C = \angle B$, we have $\angle C = 50^\circ$.

Final Answer: $\angle B = 50^\circ$ and $\angle C = 50^\circ$.

To Find: The value of the missing angles in each figure.

Given: $\triangle \mathrm{HEN} \cong \triangle \mathrm{BIG}$

This congruence implies the following correspondence between vertices: H $\leftrightarrow$ B E $\leftrightarrow$ I N $\leftrightarrow$ G

To list all correct ways of expressing this congruence, we must maintain this correspondence while changing the order of the vertices.

Given: Two triangles with their side lengths specified.

Triangle 1: Let's name it $\triangle ABC$. The side lengths are AB = 3, BC = 5, and AC = 4.

Triangle 2: Let's name it $\triangle PQR$. The side lengths are PQ = 3, QR = 5, and PR = 4.

To Determine: Whether $\triangle ABC$ is congruent to $\triangle PQR$.

Solution: We can compare the side lengths of the two triangles.

Since the three sides of $\triangle ABC$ are equal to the three corresponding sides of $\triangle PQR$, the triangles are congruent by the SSS (Side-Side-Side) congruence condition.

Expressing the Congruence: The correspondence of vertices is A $\leftrightarrow$ P, B $\leftrightarrow$ Q, and C $\leftrightarrow$ R. Therefore, the congruence is written as: $\triangle ABC \cong \triangle PQR$

Given: A quadrilateral ABCD with a diagonal AC. We are given that AB = AD and CB = CD.

Yes, we can identify a pair of congruent triangles: $\triangle ABC$ and $\triangle ADC$.

Reason for Congruence: Let's consider $\triangle ABC$ and $\triangle ADC$.

Since the three sides of $\triangle ABC$ are equal to the corresponding three sides of $\triangle ADC$, the triangles are congruent by the SSS (Side-Side-Side) congruence condition. So, $\triangle ABC \cong \triangle ADC$.

Yes, AC divides both $\angle BAD$ and $\angle BCD$ into two equal parts. This means AC is the angle bisector for both angles.

Reason: Since we have proved that $\triangle ABC \cong \triangle ADC$, their corresponding parts must be equal. This is often referred to as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Given: Two triangles, $\triangle DFE$ and $\triangle DGE$, which share a common side DE. We are given that DF = DG and FE = GE.

To Determine: Whether $\triangle DFE \cong \triangle DGE$.

Solution: Let's analyze the two triangles, $\triangle DFE$ and $\triangle DGE$, to check for congruence conditions.

We have found that the three sides of $\triangle DFE$ are equal to the corresponding three sides of $\triangle DGE$.

Therefore, by the SSS (Side-Side-Side) congruence condition, the triangles are congruent.

Final Answer: Yes, $\triangle DFE$ and $\triangle DGE$ are congruent to each other.

This question involves analyzing two separate pairs of triangles.

Given: A figure with a quadrilateral ABCD and a diagonal BD. We are given AB || CD and AB = CD.

To Find: Other equal parts and to check if $\triangle ABD$ and $\triangle CDB$ are congruent.

Final Answer: The triangles are congruent: $\triangle ABD \cong \triangle CDB$. The other equal parts are: AD = CB, $\angle DAB = \angle BCD$, and $\angle ADB = \angle CBD$.

Given: Two triangles, $\triangle ABC$ and $\triangle DBC$, sharing a common side BC. We are given $\angle ABC = \angle DBC$ and $\angle ACB = \angle DCB$.