Question

if $delta bat cong delta dog$, and $mangle b = 14^circ$, $mangle g = 29^circ$ and $mangle o = (10x + 7)^circ$, find $x$ and $mangle o$.

Explanation:

Step1: Match congruent triangle angles

Since $\triangle BAT \cong \triangle DOG$, corresponding angles are equal:
$m\angle A = m\angle O$, $m\angle B = m\angle D = 14^\circ$, $m\angle T = m\angle G = 29^\circ$

Step2: Sum of triangle angles is $180^\circ$

Calculate $m\angle O$ using angle sum property:
$m\angle D + m\angle O + m\angle G = 180^\circ$
$14^\circ + (10x+7)^\circ + 29^\circ = 180^\circ$

Step3: Simplify and solve for $x$

Combine constants first:
$14 + 7 + 29 + 10x = 180$
$50 + 10x = 180$
$10x = 180 - 50$
$10x = 130$
$x = \frac{130}{10} = 13$

Step4: Calculate $m\angle O$

Substitute $x=13$ into the expression:
$m\angle O = (10\times13 + 7)^\circ = 137^\circ$

Answer:

$x = 13$
$m\angle O = 137^\circ$