Question

Observe the information shown in pairs of triangles given below. State the test by which the two triangles are congruent. Write the remaining congruent parts of the triangles.

(i)

From the information shown in the figure,

in $∆$ABC and $∆$PQR

$\angle$ABC $\cong \angle$PQR

seg BC $\cong$ seg QR

$\angle$ACB$\cong$ $\angle$PRQ

$\therefore ∆$ABC $\cong$ $∆$PQR ....... $\boxed{3243432}$ test

$\therefore \angle$BAC $\cong$ $\boxed{3243432}$ .......corresponding angles of congruent triangles.

$\left.\begin{array}{l}\mathrm{seg}\mathrm{AB}\cong \boxed{1234}\\ \mathrm{and}\boxed{1234}\cong \mathrm{seg}\mathrm{PR}\end{array}\right\}...\mathrm{corresponding}\mathrm{sides}\mathrm{of}\mathrm{congruent}\mathrm{triangles}$


(ii)

From the information shown in the figure,

in $∆$PTQ and $∆$STR

seg PT $\cong$ seg ST

$\angle$PTQ$\cong$ $\angle$STR...... vertically opposite angles
$\therefore ∆$PTQ $\cong$ $∆$STR ....... $\boxed{3243432}$ test

$\left.\begin{array}{l}\therefore \angle \mathrm{TPQ}\cong \boxed{1234}\\ \mathrm{and}\boxed{1234}\cong \angle \mathrm{TRS}\end{array}\right\}...\mathrm{corresponding}\mathrm{angles}\mathrm{of}\mathrm{congruent}\mathrm{triangles}$

seg PQ $\cong$ $\boxed{3243432}$ corresponding sides of congruent triangles

Answer 1

(i)

From the information shown in the figure,

in $∆$ABC and $∆$PQR

$\angle$ABC $\cong \angle$PQR

seg BC $\cong$ seg QR

$\angle$ACB$\cong$ $\angle$PRQ

$\therefore ∆$ABC $\cong$ $∆$PQR ....... $\boxed{\mathrm{ASA}}$ test

$\therefore \angle$BAC $\cong$ $\boxed{\angle \mathrm{QPR}}$ .......corresponding angles of congruent triangles.

$\left.\begin{array}{l}\mathrm{seg}\mathrm{AB}\cong \boxed{\mathrm{seg}\mathrm{PQ}}123\mathrm{A}\\ \mathrm{and}\boxed{\mathrm{seg}\mathrm{AC}}\cong \mathrm{seg}\mathrm{PR}\end{array}\right\}...\mathrm{corresponding}\mathrm{sides}\mathrm{of}\mathrm{congruent}\mathrm{triangles}$


(ii)

From the information shown in the figure,

in $∆$PTQ and $∆$STR

seg PT $\cong$ seg ST

$\angle$PTQ$\cong$ $\angle$STR...... vertically opposite angles
$\therefore ∆$PTQ $\cong$ $∆$STR ....... $\boxed{\mathrm{SAS}}$ test

$\left.\begin{array}{l}\therefore \angle \mathrm{TPQ}\cong \boxed{\angle \mathrm{TSR}}\\ \mathrm{and}\boxed{\angle \mathrm{TQP}}\cong \angle \mathrm{TRS}\end{array}\right\}...\mathrm{corresponding}\mathrm{angles}\mathrm{of}\mathrm{congruent}\mathrm{triangles}$

seg PQ $\cong$ $\boxed{\mathrm{seg}\mathrm{SR}}$ corresponding sides of congruent triangles