$ l $ and $ m $ are two parallel lines intersected by another pair of parallel lines $ p $ and $ q $ (see Fig. 7.19). Show that $ \triangle \mathrm{ABC} \cong \triangle \mathrm{CDA} $. "
Given:
$l$ and $m$ are two parallel lines intersected by another pair of parallel lines $p$ and $q$.
To do:
We have to show that $\triangle ABC\cong \triangle CDA$.
Solution:
Let us consider $\triangle ABC$ and $\triangle CDA$,
We know that,
When the lines intersected by the transversal are parallel, alternate interior angles are equal.
This implies,
$\angle BCA=\angle DAC$ and $BAC=\angle DCA$
Since $AC$ and $CA$ is the common side of the triangles,
We get,
$AC=CA$
Therefore,
From ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.
We get,
$\triangle ABC \cong\triangle CDA$.
Related Articles Two sides \( \mathrm{AB} \) and \( \mathrm{BC} \) and median \( \mathrm{AM} \) of one triangle \( \mathrm{ABC} \) are respectively equal to sides \( \mathrm{PQ} \) and \( \mathrm{QR} \) and median \( \mathrm{PN} \) of \( \triangle \mathrm{PQR} \) (see Fig. 7.40). Show that:(i) \( \triangle \mathrm{ABM} \equiv \triangle \mathrm{PQN} \)(ii) \( \triangle \mathrm{ABC} \cong \triangle \mathrm{PQR} \)"\n
\( \mathrm{ABC} \) is a triangle in which altitudes \( \mathrm{BE} \) and \( \mathrm{CF} \) to sides \( \mathrm{AC} \) and \( \mathrm{AB} \) are equal (see Fig. 7.32). Show that(i) \( \triangle \mathrm{ABE} \cong \triangle \mathrm{ACF} \)(ii) \( \mathrm{AB}=\mathrm{AC} \), i.e., \( \mathrm{ABC} \) is an isosceles triangle."\n
In \( \triangle \mathrm{ABC}, \mathrm{AD} \) is the perpendicular bisector of \( \mathrm{BC} \) (see Fig. 7.30). Show that \( \triangle \mathrm{ABC} \) is an isosceles triangle in which \( \mathrm{AB}=\mathrm{AC} \)."\n
In quadrilateral \( A C B D \),\( \mathrm{AC}=\mathrm{AD} \) and \( \mathrm{AB} \) bisects \( \angle \mathrm{A} \) (see Fig. 7.16). Show that \( \triangle \mathrm{ABC} \cong \triangle \mathrm{ABD} \).What can you say about \( \mathrm{BC} \) and \( \mathrm{BD} \) ?"64391"\n
If $l, m, n$ are three lines such that $l \parallel m$ and $n \perp l$, prove that $n \perp m$.
In the figure, $l, m$ and $n$ are parallel lines intersected by transversal $p$ at $X, Y$ and $Z$ respectively. Find $\angle l, \angle 2$ and $\angle 3$."\n
In Fig. 7.48, sides \( \mathrm{AB} \) and \( \mathrm{AC} \) of \( \triangle \mathrm{ABC} \) are extended to points \( \mathrm{P} \) and \( \mathrm{Q} \) respectively. Also, \( \angle \mathrm{PBC}\mathrm{AB} \)."\n
\( \mathrm{AB} \) is a line segment and \( \mathrm{P} \) is its mid-point. \( \mathrm{D} \) and \( \mathrm{E} \) are points on the same side of \( \mathrm{AB} \) such that \( \angle \mathrm{BAD}=\angle \mathrm{ABE} \) and \( \angle \mathrm{EPA}=\angle \mathrm{DPB} \) (see Fig. 7.22). Show that(i) \( \triangle \mathrm{DAP} \cong \triangle \mathrm{EBP} \)(ii) \( \mathrm{AD}=\mathrm{BE} \)"\n
\( \mathrm{ABC} \) is an isosceles triangle in which altitudes \( \mathrm{BE} \) and \( \mathrm{CF} \) are drawn to equal sides \( \mathrm{AC} \) and \( \mathrm{AB} \) respectively (see Fig. 7.31). Show that these altitudes are equal."\n
\( \triangle \mathrm{ABC} \) and \( \triangle \mathrm{DBC} \) are two isosceles triangles on the same base \( B C \) and vertices \( A \) and \( D \) are on the same side of \( \mathrm{BC} \) (see Fig. 7.39). If \( \mathrm{AD} \) is extended to intersect \( \mathrm{BC} \) at \( \mathrm{P} \), show that(i) \( \triangle \mathrm{ABD} \equiv \triangle \mathrm{ACD} \)(ii) \( \triangle \mathrm{ABP} \cong \triangle \mathrm{ACP} \)(iii) \( \mathrm{AP} \) bisects \( \angle \mathrm{A} \) as well as \( \angle \mathrm{D} \).(iv) AP is the perpendicular bisector of \( \mathrm{BC} \)."
\( \mathrm{ABCD} \) is a parallelogram and \( \mathrm{AP} \) and \( \mathrm{CQ} \) are perpendiculars from vertices \( \mathrm{A} \) and \( \mathrm{C} \) on diagonal \( \mathrm{BD} \) (see below figure). Show that(i) \( \triangle \mathrm{APB} \cong \triangle \mathrm{CQD} \)(ii) \( \mathrm{AP}=\mathrm{CQ} \)"\n
In parallelogram \( \mathrm{ABCD} \), two points \( \mathrm{P} \) and \( \mathrm{Q} \) are taken on diagonal \( \mathrm{BD} \) such that \( \mathrm{DP}=\mathrm{BQ} \) (see below figure). Show that:(i) \( \triangle \mathrm{APD} \equiv \triangle \mathrm{CQB} \)(ii) \( \mathrm{AP}=\mathrm{CQ} \)(iii) \( \triangle \mathrm{AQB} \equiv \triangle \mathrm{CPD} \)(iv) \( \mathrm{AQ}=\mathrm{CP} \)(v) \( \mathrm{APCQ} \) is a parallelogram"\n
\( \mathrm{ABC} \) and \( \mathrm{DBC} \) are two isosceles triangles on the same base \( BC \) (see Fig. 7.33). Show that \( \angle \mathrm{ABD}=\angle \mathrm{ACD} \)."\n
$ABC$ is a triangle and through $A, B, C$ lines are drawn parallel to $BC, CA$ and $AB$ respectively intersecting at $P, Q$ and $R$. Prove that the perimeter of $\triangle PQR$ is double the perimeter of $\triangle ABC$.
In \( \triangle \mathrm{ABC}, \mathrm{M} \) and \( \mathrm{N} \) are the midpoints of \( \mathrm{AB} \) and \( \mathrm{AC} \) respectively. If the area of \( \triangle \mathrm{ABC} \) is \( 90 \mathrm{~cm}^{2} \), find the area of \( \triangle \mathrm{AMN} \).
Kickstart Your Career
Get certified by completing the course
Get Started