3Number 3ExplanationLines and AnglesTwo parallel lines PQPQPQ and RSRSRS cut by a zig-zag line forming angles 25∘25^\circ25∘, 135∘135^\circ135∘, and α\alphaα.LoadingToggle GridToggle Play If PQPQPQ is parallel to RSRSRS, then the value of α\alphaα is...45∘45^\circ45∘65∘65^\circ65∘70∘70^\circ70∘75∘75^\circ75∘80∘80^\circ80∘ExplanationSolution VisualizationAuxiliary line divides angle α\alphaα into β\betaβ and γ\gammaγ.LoadingToggle GridToggle Play We draw an auxiliary line parallel to lines PQPQPQ and RSRSRS passing through the vertex of angle α\alphaα. This line divides angle α\alphaα into two parts, β\betaβ and γ\gammaγ. Finding the Value of β\betaβ Notice that angle β\betaβ and the 25∘25^\circ25∘ angle are alternate interior angles. Since the lines are parallel, their measures are equal: β=25∘\beta = 25^\circβ=25∘ Finding the Value of γ\gammaγ Angle γ\gammaγ and the 135∘135^\circ135∘ angle are consecutive interior angles. The sum of these two angles is 180∘180^\circ180∘: γ+135∘=180∘\gamma + 135^\circ = 180^\circγ+135∘=180∘γ=180∘−135∘=45∘\gamma = 180^\circ - 135^\circ = 45^\circγ=180∘−135∘=45∘ Calculating the Value of α\alphaα The value of α\alphaα is the sum of β\betaβ and γ\gammaγ: α=β+γ\alpha = \beta + \gammaα=β+γα=25∘+45∘\alpha = 25^\circ + 45^\circα=25∘+45∘α=70∘\alpha = 70^\circα=70∘ Therefore, the value of α\alphaα is 70∘70^\circ70∘.
3Number 3ExplanationLines and AnglesTwo parallel lines PQPQPQ and RSRSRS cut by a zig-zag line forming angles 25∘25^\circ25∘, 135∘135^\circ135∘, and α\alphaα.LoadingToggle GridToggle Play If PQPQPQ is parallel to RSRSRS, then the value of α\alphaα is...45∘45^\circ45∘65∘65^\circ65∘70∘70^\circ70∘75∘75^\circ75∘80∘80^\circ80∘ExplanationSolution VisualizationAuxiliary line divides angle α\alphaα into β\betaβ and γ\gammaγ.LoadingToggle GridToggle Play We draw an auxiliary line parallel to lines PQPQPQ and RSRSRS passing through the vertex of angle α\alphaα. This line divides angle α\alphaα into two parts, β\betaβ and γ\gammaγ. Finding the Value of β\betaβ Notice that angle β\betaβ and the 25∘25^\circ25∘ angle are alternate interior angles. Since the lines are parallel, their measures are equal: β=25∘\beta = 25^\circβ=25∘ Finding the Value of γ\gammaγ Angle γ\gammaγ and the 135∘135^\circ135∘ angle are consecutive interior angles. The sum of these two angles is 180∘180^\circ180∘: γ+135∘=180∘\gamma + 135^\circ = 180^\circγ+135∘=180∘γ=180∘−135∘=45∘\gamma = 180^\circ - 135^\circ = 45^\circγ=180∘−135∘=45∘ Calculating the Value of α\alphaα The value of α\alphaα is the sum of β\betaβ and γ\gammaγ: α=β+γ\alpha = \beta + \gammaα=β+γα=25∘+45∘\alpha = 25^\circ + 45^\circα=25∘+45∘α=70∘\alpha = 70^\circα=70∘ Therefore, the value of α\alphaα is 70∘70^\circ70∘.
